{"id":1043,"date":"2013-06-01T11:37:38","date_gmt":"2013-06-01T17:37:38","guid":{"rendered":"http:\/\/www.ghyzmo.com\/?p=1043"},"modified":"2013-06-30T14:27:07","modified_gmt":"2013-06-30T20:27:07","slug":"the-hydrogen-atom-analyzed-using-commutator-algebra-i","status":"publish","type":"post","link":"http:\/\/www.ghyzmo.com\/the-hydrogen-atom-analyzed-using-commutator-algebra-i\/","title":{"rendered":"The Hydrogen Atom Analyzed Using Commutator Algebra I"},"content":{"rendered":"

\u00a9Fernando Caracena<\/em> 2013<\/p>\n

On a previous blog on the one-dimensional harmonic oscillator, we showed that the algebra of non-commuting operators in quantum mechanics could be exploited systematically to solve for the entire spectrum of state vectors and energy eigenvalues. Algebra sufficed for most of the calculation. In the following series we will develop the techniques of Lie algebra<\/a> using non-commuting operators to solve for the state vectors and corresponding eigenvalues for the electrons in a hydrogen atom. In doing so, we will cover some techniques for systematically solving problems that have several, simultaneous eigenvalues. In three dimensional problems such a that of atomic structure, angular momentum plays a role in organizing the motion of electrons. We will find that angular momentum is the source of two additional eigenvalues in addition to that of energy.<\/p>\n

The Problem of the Hydrogen Atom<\/h2>\n

The electron moving in a stationary state in a hydrogen atom satisfies the time independent Schr\u00f6dinger Equation,
\nH<\/strong>|\u03a6> =E<\/span><\/span>n<\/span><\/span><\/sub> |\u03a6>,\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 (1a)
\n<\/strong>where the Hamiltonian operator is the following
\n<\/strong>H<\/strong> = p<\/strong>2<\/span><\/sup>\/2m\u00a0\u2013 k e2<\/span><\/sup>\/R<\/strong>2<\/span><\/sup>,\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 (1b)
\n<\/strong>and E<\/span><\/span>n<\/span><\/span><\/sub>
\n<\/strong>are its eigenvalues.<\/p>\n

Preliminaries Old Quantum Theory Solution<\/a><\/h3>\n

Bohr solved for the energy eigenvalues as follows by using the old quantum theory, which are negative because they represent bound states of the electron:
\n<\/strong>En<\/sub> = \u2013\u00bd (k e2<\/sup>)2<\/sup>me<\/sub>\/\u01272<\/sup> (1\/n2<\/sup>).\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 (1c)
\n<\/strong>[Free electron states are represented by positive values.]
\n<\/strong>The constant k is given by
\n<\/strong>k=1\/(4 \u03c0\u03b50<\/sub>).
\n<\/strong>The expression for energy eigenvalues can be simplified by writing it as
\n<\/strong>En<\/sub> = -\u00bd k e2<\/sup>\/a0<\/sub> (1\/n2<\/sup>),\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 (1d)
\n<\/strong>where\u00a0 a0<\/sub> is known as the Bohr Radius
\n<\/strong>a0<\/sub> = \u0127\/(k e2<\/sup> me<\/sub>)
\n<\/strong>or
\n<\/strong>a0<\/sub> = 4\u03c0\u03b50<\/sub>\u0127\/(e2<\/sup> me<\/sub>).\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 (1e)
\n<\/strong>Modern quantum theory finds the same energy eigenvalues, but finds that there are two more eigenvalues associated with the angular momentum of the orbital electron.<\/p>\n

Eigenvalues and Casimir Operators<\/h2>\n

<\/strong>Casimir operators are those of the group of operators that commute among themselves, that is their order of multiplication does not matter. Commutation relations play an important role in modern quantum theory. They are a calculational tools that reduce much of the\u00a0 mathematics of quantum mechanics to algebra.<\/p>\n

Commutation relations are written for operators, some of which do not necessarily commute, that is, their product depends on the order that it is taken, such as in the following, in which the commutator is defined as
\n<\/strong>[A<\/strong>, B<\/strong>]= A<\/strong> B<\/strong>\u00a0\u2013 B<\/strong> A<\/strong>.\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 (2a)
\n<\/strong>If order does not matter, such as between X<\/strong> and Y<\/strong>, then
\n<\/strong>[X<\/strong>,Y<\/strong>]=0,
\n<\/strong>their commutator is zero.<\/p>\n

Note that we write quantum operators in upper case, bold letters.
\n<\/strong>The following is a very useful theorem that we will use repeatedly
\n<\/strong>[A <\/strong>B<\/strong>, C<\/strong>]= [A<\/strong>, C<\/strong>] B + <\/strong>A<\/strong><\/strong> [B<\/strong>, C<\/strong>].<\/strong>\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 (2b)
\n<\/strong>Note that the proof falls out just by writing out the operators
\n<\/strong>A B<\/span><\/span><\/span><\/span><\/span><\/span><\/strong><\/strong>C\u00a0\u2013<\/strong> C A B = <\/span><\/span><\/span><\/span><\/span><\/span><\/strong>A<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/strong><\/strong>C<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/strong><\/strong>B<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/strong>\u00a0\u2013<\/strong> C A B + A B<\/span><\/span><\/span><\/span><\/span><\/span><\/strong><\/strong>C\u00a0\u2013<\/strong> <\/span><\/span><\/span><\/span><\/span><\/span><\/strong>A C B.<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/strong>
\n<\/strong>Another almost trivial theorem is left to the reader to verify
\n<\/strong>[A<\/strong>, B<\/strong>]=\u2013<\/strong>[B<\/strong>, A<\/strong>].\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 (2c)<\/p>\n

Heisenberg's Uncertainty relations apply to expectation values of noncommuting opertors.<\/p>\n

If operators, such as X<\/strong> and P<\/strong>, fail to commute, their values cannot be defined simultaneously with arbitrary precision. For example, Heisenberg's Uncertainty relations apply to components of position and corresponding components of momentum<\/p>\n

\u0394x \u0394px<\/sub> \u2265 \u0127\/2<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/strong> ,\u00a0 \u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 \u00a0 (3a)
\n\u0394y \u0394p<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/strong>y<\/span><\/span><\/span><\/span><\/span><\/sub><\/span><\/span><\/strong> \u2265 \u0127\/2<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/strong><\/strong> ,\u00a0 \u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 (3b)
\n\u0394z \u0394p<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/strong>z<\/span><\/span><\/span><\/span><\/span><\/sub><\/span><\/span><\/strong> \u2265 \u0127\/2<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/strong><\/strong> ,\u00a0 \u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 (3c)
\n<\/strong><\/p>\n

where the '\u0394's represent the root mean square (RMS) expectation values defined for x, y and z as<\/p>\n

\u0394x<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/strong>2<\/span><\/span><\/span><\/span><\/span><\/sup><\/span><\/span><\/strong> = <\u03a6| (<\/span><\/span><\/span><\/span><\/span><\/span><\/span>X <\/span><\/span><\/span><\/span><\/span><\/span>\u2013<\/strong> <\u03a6|<\/span><\/span><\/span><\/span><\/span><\/span><\/span>X<\/span><\/span><\/span><\/span><\/span><\/span>|\u03a6>)<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/strong>2<\/span><\/span><\/span><\/span><\/span><\/sup><\/span><\/span><\/strong>|\u03a6><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/strong>, \u00a0 \u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 \u00a0 (4a)
\n\u0394y<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/strong>2<\/span><\/span><\/span><\/span><\/span><\/sup><\/span><\/span><\/strong> = <\u03a6| (<\/span><\/span><\/span><\/span><\/span><\/span><\/span>Y <\/span><\/span><\/span><\/span><\/span><\/span>\u2013 <\u03a6|<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/strong>|\u03a6>)<\/span><\/span><\/span><\/span><\/span><\/span><\/span>2<\/span><\/span><\/span><\/span><\/span><\/sup><\/span><\/span>|\u03a6><\/span><\/span><\/span><\/span><\/span><\/span><\/span>, \u00a0 \u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 \u00a0 (4b)
\n\u0394z<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/strong>2<\/span><\/span><\/span><\/span><\/span><\/sup><\/span><\/span><\/strong> = <\u03a6| (<\/span><\/span><\/span><\/span><\/span><\/span><\/span>Z<\/span><\/span><\/span><\/span><\/span><\/span> \u2013<\/strong> <\u03a6|<\/span><\/span><\/span><\/span><\/span><\/span><\/span>Z<\/span><\/span><\/span><\/span><\/span><\/span>|\u03a6>)<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/strong>2<\/span><\/span><\/span><\/span><\/span><\/sup><\/span><\/span><\/strong>|\u03a6>.<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/strong><\/strong> \u00a0 \u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 \u00a0 (4c)
\n<\/strong><\/p>\n

Each pair of conjugate variables, such as coordinates and momentum components, are represented by operators that have the commutation relations:<\/p>\n

[X<\/strong>, Px<\/sub><\/strong>] = i \u0127<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/strong><\/strong>, \u00a0 \u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 \u00a0 (5a)
\n<\/strong>[Y<\/strong>, Py<\/sub><\/strong>] = i \u0127<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/strong><\/strong>, \u00a0 \u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 \u00a0 (5b)
\n<\/strong>[Z<\/strong>, Pz<\/sub><\/strong>] = i \u0127.<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/strong><\/strong> \u00a0 \u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 \u00a0 (5c)
\n<\/strong><\/p>\n

The Heisenberg Uncertainty relations between them result mathematically from the failure of their representative operators to commute, which is not shown here, but perhaps in a later blog.<\/p>\n

All of the quantities of a quantum mechanical system that represent variables that can be measured simultaneously with arbitrary precision are represented by operators that commute among themselves, i. e., they are Casimir operators.<\/p>\n

Some further properties involving commutators<\/h3>\n

Commutators can be used instead of derivatives, thereby reducing a lot of calculation in quantum physics to algebra. For example, (5a) is just the beginning of a series of derivatives that can be taken of the various powers and functions of X<\/span><\/span><\/span><\/span><\/span><\/span><\/strong><\/strong><\/strong><\/strong>:
\n<\/strong><\/p>\n

[<\/span><\/span><\/span><\/span><\/span><\/span><\/span>X<\/span><\/span><\/span><\/span><\/span><\/span><\/strong><\/strong>2<\/sup> , <\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/strong>P<\/span><\/span><\/span><\/span><\/span><\/span><\/strong><\/strong>x<\/sub><\/strong><\/span><\/span><\/span><\/span><\/sub><\/span><\/span><\/strong>] = i \u0127<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/strong><\/strong>(2<\/span><\/span><\/span><\/span><\/span><\/span><\/span>X<\/span><\/span><\/span><\/span><\/span><\/span><\/strong>) = i \u0127 d(<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/strong>X<\/span><\/span><\/span><\/span><\/span><\/span><\/strong><\/strong><\/strong>2<\/sup><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/strong>)\/dX\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0<\/strong><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/strong>\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 (6a)<\/span><\/span><\/span><\/span><\/span><\/span><\/span>
\n<\/strong>[<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/strong>X<\/span><\/span><\/span><\/span><\/span><\/span><\/strong><\/strong>n<\/sup>, <\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/strong>P<\/span><\/span><\/span><\/span><\/span><\/span><\/strong>x<\/sub><\/strong><\/span><\/span><\/span><\/span><\/span><\/span><\/strong><\/strong>] = i <\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/strong><\/strong>\u0127 <\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/strong>n<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/strong><\/strong>X<\/span><\/span><\/span><\/span><\/span><\/span><\/strong><\/strong>n-1<\/sup><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/strong>\u00a0 =<\/strong>i \u0127<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/strong> d<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/strong>X<\/span><\/span><\/span><\/span><\/span><\/span><\/strong><\/strong><\/strong><\/strong><\/strong>n<\/sup><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/strong><\/sup><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/strong>\/dX\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0<\/strong><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/strong>\u00a0\u00a0 (6b)<\/span><\/span><\/span><\/span><\/span><\/span><\/span>
\n<\/strong>[<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/strong>f(<\/span><\/span><\/span><\/span><\/span><\/span>X<\/span><\/span><\/span><\/span><\/span><\/span><\/strong><\/strong>)<\/strong>, <\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/strong>P<\/span><\/span><\/span><\/span><\/span><\/span><\/strong>x<\/sub><\/strong><\/span><\/span><\/span><\/span><\/sub><\/span><\/span><\/strong>] = <\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/strong>i \u0127 df(X)\/dX.<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/strong>\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0<\/strong><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/strong>\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 (6c)<\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 <\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/strong>
\n<\/strong><\/address>\n
Note that (6a) follows directly by applying (2b) then (5a). The reader can prove (6b) by mathematical induction, and (6c) by expanding f(X<\/strong>) in a Taylor Series in powers of X<\/strong>. <\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/strong><\/address>\n

\n<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/strong><\/address>\n

Identifying the Casimir operators for the hydrogen atom<\/em><\/h2>\n

The energy operator, or Hamiltonian, is a candidate for one of the Casimir operators for the hydrogen atom. What are some other possibilities? A good place to look is at angular momentum, which as an operator is defined as follows
\n<\/strong>L<\/strong> = R<\/strong> x P<\/strong>.
\n<\/strong>The components of the angular momentum vector components (L<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/strong>x<\/span><\/span><\/span><\/span><\/sub><\/span><\/span><\/span><\/strong><\/strong><\/strong>, L<\/span><\/span><\/span><\/span><\/span><\/span><\/strong>y, <\/span><\/span><\/span><\/span><\/sub><\/span><\/span><\/strong>L<\/span><\/span><\/span><\/span><\/span><\/span><\/strong>z<\/span><\/span><\/span><\/span><\/sub><\/span><\/span><\/strong><\/strong><\/strong>) are each the following operators:
\n<\/strong>L<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/strong>x<\/span><\/span><\/span><\/span><\/sub><\/span><\/span><\/span><\/strong><\/strong>= <\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/strong>Y<\/span><\/strong><\/span><\/span><\/span><\/span><\/span><\/span><\/strong><\/strong>P<\/span><\/strong><\/span><\/span><\/span><\/span><\/span><\/span><\/strong>z<\/span><\/span><\/span><\/span><\/span><\/sub><\/span><\/span><\/span><\/strong> \u2013 <\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/strong>Z<\/span><\/strong><\/span><\/span><\/span><\/span><\/span><\/span><\/strong><\/strong>P<\/span><\/strong><\/span><\/span><\/span><\/span><\/span><\/span><\/strong>y<\/span><\/span><\/span><\/span><\/span><\/sub><\/span><\/span><\/span><\/strong>,<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/strong> \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0\u00a0 \u00a0 (7a)
\nL<\/span><\/span><\/span><\/span><\/span><\/span><\/strong>y<\/span><\/span><\/span><\/span><\/sub><\/span><\/span><\/strong><\/strong>= <\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/strong>Z<\/span><\/strong><\/span><\/span><\/span><\/span><\/span><\/strong><\/strong>P<\/span><\/strong><\/span><\/span><\/span><\/span><\/span><\/strong>x<\/span><\/span><\/span><\/span><\/span><\/sub><\/span><\/span><\/strong> \u2013 <\/span><\/span><\/span><\/strong>X<\/span><\/strong><\/span><\/span><\/span><\/span><\/span><\/strong><\/strong>P<\/span><\/strong><\/span><\/span><\/span><\/span><\/span><\/strong>z<\/span><\/span><\/span><\/span><\/span><\/sub><\/span><\/span><\/strong>,<\/strong> \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0\u00a0 \u00a0 (7b)
\nL<\/span><\/span><\/span><\/span><\/span><\/span><\/strong>z<\/span><\/span><\/span><\/span><\/sub><\/span><\/span><\/strong><\/strong>= <\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/strong>X<\/span><\/strong><\/span><\/span><\/span><\/span><\/span><\/strong><\/strong>P<\/span><\/strong><\/span><\/span><\/span><\/span><\/span><\/strong>y<\/span><\/span><\/span><\/span><\/span><\/sub><\/span><\/span><\/strong> \u2013 <\/span><\/span><\/span><\/strong>Y<\/span><\/strong><\/span><\/span><\/span><\/span><\/span><\/strong><\/strong>P<\/span><\/strong><\/span><\/span><\/span><\/span><\/span><\/strong>x<\/span><\/span><\/span><\/span><\/span><\/sub><\/span><\/span><\/strong> .<\/strong> \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0\u00a0\u00a0 (7c)<\/p>\n

Note that the components defined in (7a), (7b) and (7c) do ont commute among themselves, which we will show for L<\/span><\/span><\/span><\/span><\/span><\/span><\/strong>x<\/span><\/span><\/span><\/span><\/sub><\/span><\/span><\/strong>L<\/span><\/span><\/span><\/span><\/span><\/span><\/strong>y<\/span><\/span><\/span><\/span><\/sub><\/span><\/span><\/strong>:
\n<\/strong>[L<\/span><\/span><\/span><\/span><\/span><\/span><\/strong>x<\/span><\/span><\/span><\/span><\/sub><\/span><\/span><\/strong>, <\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/strong>L<\/span><\/span><\/span><\/span><\/span><\/span><\/strong>y<\/span><\/span><\/span><\/span><\/sub><\/span><\/span><\/strong>] = Y<\/span><\/strong><\/span><\/span><\/span><\/span><\/span><\/strong><\/strong>[P<\/span><\/strong><\/span><\/span><\/span><\/span><\/span><\/strong>z<\/span><\/span><\/span><\/span><\/span><\/sub><\/span><\/span><\/strong>, <\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/strong>Z<\/span><\/strong><\/span><\/span><\/span><\/span><\/span><\/strong>] <\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/strong>P<\/span><\/strong><\/span><\/span><\/span><\/span><\/span><\/strong>x<\/span><\/span><\/span><\/span><\/span><\/sub><\/span><\/span><\/strong><\/strong>+ <\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/strong>P<\/span><\/strong><\/span><\/span><\/span><\/span><\/span><\/strong>y<\/span><\/span><\/span><\/span><\/span><\/sub><\/span><\/span><\/strong><\/strong>X<\/span><\/strong><\/span><\/span><\/span><\/span><\/span><\/strong><\/strong>[Z, P<\/span><\/strong><\/span><\/span><\/span><\/span><\/span><\/strong>z<\/span><\/span><\/span><\/span><\/span><\/sub><\/span><\/span><\/strong>]<\/span><\/strong><\/span><\/span><\/span><\/span><\/span><\/strong>\u00a0\u00a0 ;
\n<\/strong>[L<\/span><\/span><\/span><\/span><\/span><\/span><\/strong>x<\/span><\/span><\/span><\/span><\/sub><\/span><\/span><\/strong>, <\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/strong>L<\/span><\/span><\/span><\/span><\/span><\/span><\/strong>y<\/span><\/span><\/span><\/span><\/sub><\/span><\/span><\/strong>] = -i \u0127 <\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/strong>Y<\/span><\/strong><\/span><\/span><\/span><\/span><\/span><\/strong><\/strong>P<\/span><\/strong><\/span><\/span><\/span><\/span><\/span><\/strong>x<\/span><\/span><\/span><\/span><\/span><\/sub><\/span><\/span><\/strong><\/strong>+ i <\/span><\/strong><\/span><\/span><\/span><\/span><\/span><\/strong>\u0127 <\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/strong>P<\/span><\/strong><\/span><\/span><\/span><\/span><\/span><\/strong>y<\/span><\/span><\/span><\/span><\/span><\/sub><\/span><\/span><\/strong><\/strong>X<\/span><\/strong><\/span><\/span><\/span><\/span><\/span><\/strong>\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 \u00a0 \u00a0 ;
\n<\/strong><\/strong>[L<\/span><\/span><\/span><\/span><\/span><\/span><\/strong>x<\/span><\/span><\/span><\/span><\/sub><\/span><\/span><\/strong>, <\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/strong>L<\/span><\/span><\/span><\/span><\/span><\/span><\/strong>y<\/span><\/span><\/span><\/span><\/sub><\/span><\/span><\/strong>] = i <\/span><\/strong><\/span><\/span><\/span><\/span><\/span><\/strong>\u0127 <\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/strong>[<\/span><\/span><\/span><\/span><\/span><\/span>X<\/span><\/strong><\/span><\/span><\/span><\/span><\/span><\/strong><\/strong>P<\/span><\/strong><\/span><\/span><\/span><\/span><\/span><\/strong>y<\/span><\/span><\/span><\/span><\/span><\/sub><\/span><\/span><\/strong><\/strong>-<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/strong><\/strong>Y<\/span><\/strong><\/span><\/span><\/span><\/span><\/span><\/strong><\/strong>P<\/span><\/strong><\/span><\/span><\/span><\/span><\/span><\/strong>x<\/span><\/span><\/span><\/span><\/span><\/sub><\/span><\/span><\/strong>]\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 <\/strong>;
\n<\/strong><\/strong>or<\/strong>
\n<\/strong><\/strong>[L<\/span><\/span><\/span><\/span><\/span><\/span><\/strong>x<\/span><\/span><\/span><\/span><\/sub><\/span><\/span><\/strong>, <\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/strong>L<\/span><\/span><\/span><\/span><\/span><\/span><\/strong>y<\/span><\/span><\/span><\/span><\/sub><\/span><\/span><\/strong>] = i <\/span><\/strong><\/span><\/span><\/span><\/span><\/span><\/strong>\u0127 <\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/strong>L<\/span><\/span><\/span><\/span><\/span><\/span><\/strong>z<\/span><\/span><\/span><\/span><\/sub><\/span><\/span><\/strong>\u00a0 . \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0\u00a0 (8a)<\/p>\n

\u00a0It is left as an exercise for the reader to derive the following two commutation relations:<\/strong>
\n<\/strong><\/strong>[L<\/span><\/span><\/span><\/span><\/span><\/span><\/strong>y<\/span><\/span><\/span><\/span><\/sub><\/span><\/span><\/strong>, <\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/strong>L<\/span><\/span><\/span><\/span><\/span><\/span><\/strong>z<\/span><\/span><\/span><\/span><\/sub><\/span><\/span><\/strong>] = i <\/span><\/strong><\/span><\/span><\/span><\/span><\/span><\/strong>\u0127 <\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/strong>L<\/span><\/span><\/span><\/span><\/span><\/span><\/strong>x<\/span><\/span><\/span><\/span><\/sub><\/span><\/span><\/strong><\/strong> ,\u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0\u00a0 (8b)
\n<\/strong>[L<\/span><\/span><\/span><\/span><\/span><\/span><\/strong>z<\/span><\/span><\/span><\/span><\/sub><\/span><\/span><\/strong>, <\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/strong>L<\/span><\/span><\/span><\/span><\/span><\/span><\/strong>x<\/span><\/span><\/span><\/span><\/sub><\/span><\/span><\/strong>] = <\/span><\/span><\/span><\/span><\/span><\/span><\/strong>i <\/span><\/strong><\/span><\/span><\/span><\/span><\/span><\/strong>\u0127 <\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/strong>L<\/span><\/span><\/span><\/span><\/span><\/span><\/strong>y<\/span><\/span><\/span><\/span><\/sub><\/span><\/span><\/strong> .\u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0\u00a0 (8c)<\/strong><\/strong><\/strong>
\n<\/strong><\/strong><\/strong><\/strong><\/p>\n

Note that each component commutes with itself,
\n<\/strong>[L<\/span><\/span><\/span><\/span><\/span><\/span><\/strong>x<\/span><\/span><\/span><\/span><\/sub><\/span><\/span><\/strong>, <\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/strong>L<\/span><\/span><\/span><\/span><\/span><\/span><\/strong>x<\/span><\/span><\/span><\/span><\/sub><\/span><\/span><\/strong>] =0\u00a0 ,\u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 (8d)<\/strong><\/strong><\/strong>
\n[L<\/span><\/span><\/span><\/span><\/span><\/span><\/strong>y<\/span><\/span><\/span><\/span><\/sub><\/span><\/span><\/strong>, <\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/strong>L<\/span><\/span><\/span><\/span><\/span><\/span><\/strong>y<\/span><\/span><\/span><\/span><\/sub><\/span><\/span><\/strong><\/strong>]= 0\u00a0 ,\u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 (8e)<\/strong><\/strong><\/strong>
\n[L<\/span><\/span><\/span><\/span><\/span><\/span><\/strong>z<\/span><\/span><\/span><\/span><\/sub><\/span><\/span><\/strong>, <\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/strong>L<\/span><\/span><\/span><\/span><\/span><\/span><\/strong>z<\/span><\/span><\/span><\/span><\/sub><\/span><\/span><\/strong><\/strong>]= 0\u00a0 .\u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 (8f)<\/strong><\/strong><\/strong>
\nIt is left as an exercise for the reader to show why.
\n<\/strong><\/p>\n

Nest we show that L<\/span><\/span><\/span><\/span><\/span><\/span><\/strong>z<\/span><\/span><\/span><\/span><\/sub><\/span><\/span><\/strong> commutes with <\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/strong>L<\/span><\/span><\/span><\/span><\/span><\/span><\/strong><\/strong>2<\/span><\/span><\/span><\/span><\/sup><\/span><\/span><\/strong><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/strong><\/strong><\/strong><\/strong>, where<\/strong><\/strong><\/strong>
\n<\/strong><\/strong><\/strong><\/strong>L<\/span><\/span><\/span><\/span><\/span><\/span><\/strong><\/strong>2<\/span><\/span><\/span><\/span><\/sup><\/span><\/span><\/strong> = <\/span><\/span><\/span><\/span><\/span><\/span><\/strong><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/strong>L<\/span><\/span><\/span><\/span><\/span><\/span><\/strong>x<\/span><\/span><\/span><\/span><\/sub><\/span><\/span><\/strong>2<\/span><\/span><\/span><\/span><\/sup><\/span><\/span><\/strong>+<\/span><\/span><\/span><\/span><\/span><\/span><\/strong><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/strong><\/strong>L<\/span><\/span><\/span><\/span><\/span><\/span><\/strong>y<\/span><\/span><\/span><\/span><\/sub><\/span><\/span><\/strong>2<\/span><\/span><\/span><\/span><\/sup><\/span><\/span><\/strong><\/strong>+ <\/span><\/span><\/span><\/span><\/span><\/span><\/strong><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/strong>L<\/span><\/span><\/span><\/span><\/span><\/span><\/strong>z<\/span><\/span><\/span><\/span><\/sub><\/span><\/span><\/strong>2<\/span><\/span><\/span><\/span><\/sup><\/span><\/span><\/strong>. \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 (9a)<\/strong><\/strong><\/strong>
\n<\/strong><\/strong><\/strong>This is easy, by showing that <\/span><\/span><\/span><\/span><\/span><\/span>L<\/span><\/span><\/span><\/span><\/span><\/span><\/strong>z<\/span><\/span><\/span><\/span><\/sub><\/span><\/span><\/strong><\/span><\/span><\/span><\/span><\/span><\/span><\/strong>\u00a0 <\/span><\/span><\/span><\/span><\/sub><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>commutes with the square of each component:<\/span><\/span><\/span><\/span><\/span><\/span><\/strong><\/strong><\/strong>
\n<\/strong><\/strong><\/strong><\/strong><\/strong>[L<\/span><\/span><\/span><\/span><\/span><\/span><\/strong>z<\/span><\/span><\/span><\/span><\/sub><\/span><\/span><\/strong>, <\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/strong>L<\/span><\/span><\/span><\/span><\/span><\/span><\/strong>x<\/span><\/span><\/span><\/span><\/sub><\/span><\/span><\/strong>2<\/span><\/span><\/span><\/span><\/sup><\/span><\/span><\/strong>] = [L<\/span><\/span><\/span><\/span><\/span><\/span><\/strong>z<\/span><\/span><\/span><\/span><\/sub><\/span><\/span><\/strong>, <\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/strong>L<\/span><\/span><\/span><\/span><\/span><\/span><\/strong>x<\/span><\/span><\/span><\/span><\/sub><\/span><\/span><\/strong>] L<\/span><\/span><\/span><\/span><\/span><\/span><\/strong>x<\/span><\/span><\/span><\/span><\/sub><\/span><\/span><\/strong> + L<\/span><\/span><\/span><\/span><\/span><\/span><\/strong>x<\/span><\/span><\/span><\/span><\/sub><\/span><\/span><\/strong> [L<\/span><\/span><\/span><\/span><\/span><\/span><\/strong>z<\/span><\/span><\/span><\/span><\/sub><\/span><\/span><\/strong>, <\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/strong>L<\/span><\/span><\/span><\/span><\/span><\/span><\/strong>x<\/span><\/span><\/span><\/span><\/sub><\/span><\/span><\/strong>]<\/span><\/span><\/span><\/span><\/span><\/span><\/strong> ;\u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 (9b)<\/strong><\/strong><\/strong>
\n[L<\/span><\/span><\/span><\/span><\/span><\/span><\/strong>z<\/span><\/span><\/span><\/span><\/sub><\/span><\/span><\/strong>, <\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/strong>L<\/span><\/span><\/span><\/span><\/span><\/span><\/strong>x<\/span><\/span><\/span><\/span><\/sub><\/span><\/span><\/strong>2<\/span><\/span><\/span><\/span><\/sup><\/span><\/span><\/strong>] = <\/span><\/span><\/span><\/span><\/span><\/span><\/strong>i <\/span><\/strong><\/span><\/span><\/span><\/span><\/span><\/strong>\u0127 (<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/strong>L<\/span><\/span><\/span><\/span><\/span><\/span><\/strong>y <\/span><\/span><\/span><\/span><\/sub><\/span><\/span><\/strong>L<\/span><\/span><\/span><\/span><\/span><\/span><\/strong>x <\/span><\/span><\/span><\/span><\/sub><\/span><\/span><\/strong>+ L<\/span><\/span><\/span><\/span><\/span><\/span><\/strong>x<\/span><\/span><\/span><\/span><\/sub><\/span><\/span><\/strong> L<\/span><\/span><\/span><\/span><\/span><\/span><\/strong>y<\/span><\/span><\/span><\/span><\/sub><\/span><\/span><\/strong>)<\/span><\/span><\/span><\/span><\/span><\/span><\/strong><\/strong><\/strong>\u00a0 ;\u00a0 \u00a0 \u00a0 \u00a0 \u00a0\u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 (9c)<\/strong><\/strong><\/strong>
\n<\/strong>[L<\/span><\/span><\/span><\/span><\/span><\/span><\/strong>z<\/span><\/span><\/span><\/span><\/sub><\/span><\/span><\/strong>, <\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/strong>L<\/span><\/span><\/span><\/span><\/span><\/span><\/strong>y<\/span><\/span><\/span><\/span><\/sub><\/span><\/span><\/strong>2<\/span><\/span><\/span><\/span><\/sup><\/span><\/span><\/strong>] = [L<\/span><\/span><\/span><\/span><\/span><\/span><\/strong>z<\/span><\/span><\/span><\/span><\/sub><\/span><\/span><\/strong>, <\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/strong>L<\/span><\/span><\/span><\/span><\/span><\/span><\/strong>y<\/span><\/span><\/span><\/span><\/sub><\/span><\/span><\/strong>] L<\/span><\/span><\/span><\/span><\/span><\/span><\/strong>y<\/span><\/span><\/span><\/span><\/sub><\/span><\/span><\/strong> + L<\/span><\/span><\/span><\/span><\/span><\/span><\/strong>y<\/span><\/span><\/span><\/span><\/sub><\/span><\/span><\/strong> [L<\/span><\/span><\/span><\/span><\/span><\/span><\/strong>y<\/span><\/span><\/span><\/span><\/sub><\/span><\/span><\/strong>, <\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/strong>L<\/span><\/span><\/span><\/span><\/span><\/span><\/strong>x<\/span><\/span><\/span><\/span><\/sub><\/span><\/span><\/strong>]<\/span><\/span><\/span><\/span><\/span><\/span><\/strong><\/strong><\/strong>\u00a0 \u00a0 ;\u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 (9d)<\/strong><\/strong><\/strong>
\n<\/strong>[L<\/span><\/span><\/span><\/span><\/span><\/span><\/strong>z<\/span><\/span><\/span><\/span><\/sub><\/span><\/span><\/strong>, <\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/strong>L<\/span><\/span><\/span><\/span><\/span><\/span><\/strong>y<\/span><\/span><\/span><\/span><\/sub><\/span><\/span><\/strong>2<\/span><\/span><\/span><\/span><\/sup><\/span><\/span><\/strong>] = -<\/span><\/span><\/span><\/span><\/span><\/span><\/strong>i <\/span><\/strong><\/span><\/span><\/span><\/span><\/span><\/strong>\u0127 (<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/strong> L<\/span><\/span><\/span><\/span><\/span><\/span><\/strong>x<\/span><\/span><\/span><\/span><\/sub><\/span><\/span><\/strong> L<\/span><\/span><\/span><\/span><\/span><\/span><\/strong>y <\/span><\/span><\/span><\/span><\/sub><\/span><\/span><\/strong>+ L<\/span><\/span><\/span><\/span><\/span><\/span><\/strong>y <\/span><\/span><\/span><\/span><\/sub><\/span><\/span><\/strong>L<\/span><\/span><\/span><\/span><\/span><\/span><\/strong>x <\/span><\/span><\/span><\/span><\/sub><\/span><\/span><\/strong>)<\/span><\/span><\/span><\/span><\/span><\/span><\/strong>;
\n<\/strong><\/strong><\/strong><\/strong>[L<\/span><\/span><\/span><\/span><\/span><\/span><\/strong>z<\/span><\/span><\/span><\/span><\/sub><\/span><\/span><\/strong>, <\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/strong>L<\/span><\/span><\/span><\/span><\/span><\/span><\/strong>z<\/span><\/span><\/span><\/span><\/sub><\/span><\/span><\/strong>2<\/span><\/span><\/span><\/span><\/sup><\/span><\/span><\/strong>] = 0<\/strong><\/strong><\/strong> . \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0\u00a0\u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0\u00a0\u00a0 (9e)<\/strong><\/strong><\/strong>
\n<\/strong><\/strong><\/strong><\/strong>It follows that<\/strong>
\n<\/strong><\/strong><\/strong>[L<\/span><\/span><\/span><\/span><\/span><\/span><\/strong>z<\/span><\/span><\/span><\/span><\/sub><\/span><\/span><\/strong>, <\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/strong>L<\/span><\/span><\/span><\/span><\/span><\/span><\/strong><\/strong>2<\/span><\/span><\/span><\/span><\/sup><\/span><\/span><\/strong>] =0 .<\/span><\/span><\/span><\/span><\/span><\/span><\/strong>\u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 (9f)<\/strong><\/strong><\/strong>
\n<\/strong><\/strong><\/strong>
\n<\/strong><\/strong>Finally, we will show that the radius of the orbit (R<\/strong>) commutes with all angular momentum components<\/strong>:
\n<\/strong><\/strong><\/strong> R= \u221a[X<\/span><\/strong><\/span><\/span><\/span><\/span><\/span><\/span><\/strong>2<\/span><\/span><\/span><\/span><\/span><\/sup><\/span><\/span><\/span><\/strong> + Y<\/span><\/strong><\/span><\/span><\/span><\/span><\/span><\/span><\/strong>2<\/span><\/span><\/span><\/span><\/span><\/sup><\/span><\/span><\/span><\/strong> +Z<\/span><\/strong><\/span><\/span><\/span><\/span><\/span><\/span><\/strong>2<\/span><\/span><\/span><\/span><\/span><\/sup><\/span><\/span><\/span><\/strong>]<\/span><\/strong><\/span><\/span><\/span><\/span><\/span><\/span><\/strong><\/strong> ;\u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 \u00a0 \u00a0 \u00a0 \u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 (10a)<\/strong><\/strong><\/strong>
\n<\/strong><\/strong><\/strong>\u2202<\/span><\/span><\/span><\/span><\/strong>R<\/span><\/strong><\/span><\/span><\/span><\/span><\/span><\/span><\/strong>\/\u2202<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/strong>X = X<\/span><\/strong><\/span><\/span><\/span><\/span><\/span><\/span><\/strong>\/<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/strong>R<\/span><\/strong><\/span><\/span><\/span><\/span><\/span><\/span><\/strong><\/strong><\/strong><\/strong><\/strong> ; \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 \u00a0 \u00a0 \u00a0 \u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 (10b)<\/strong><\/strong><\/strong>
\n<\/strong><\/strong><\/strong><\/strong>\u2202<\/span><\/span><\/span><\/span><\/strong>R<\/span><\/strong><\/span><\/span><\/span><\/span><\/span><\/span><\/strong>\/\u2202<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/strong>Y<\/span><\/strong><\/span><\/span><\/span><\/span><\/span><\/span><\/strong> = Y<\/span><\/strong><\/span><\/span><\/span><\/span><\/span><\/span><\/strong>\/<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/strong>R<\/span><\/strong><\/span><\/span><\/span><\/span><\/span><\/span><\/strong><\/strong><\/strong><\/strong><\/strong><\/strong> ; \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 \u00a0 \u00a0 \u00a0 \u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 (10c)<\/strong><\/strong><\/strong>
\n<\/strong><\/strong><\/strong><\/strong>\u2202<\/span><\/span><\/span><\/span><\/strong>R<\/span><\/strong><\/span><\/span><\/span><\/span><\/span><\/span><\/strong>\/\u2202<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/strong>Z<\/span><\/strong><\/span><\/span><\/span><\/span><\/span><\/span><\/strong> = Z<\/span><\/strong><\/span><\/span><\/span><\/span><\/span><\/span><\/strong>\/<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/strong>R<\/span><\/strong><\/span><\/span><\/span><\/span><\/span><\/span><\/strong><\/strong><\/strong><\/strong><\/strong><\/strong> ; \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 \u00a0 \u00a0 \u00a0 \u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 (10d)<\/strong><\/strong><\/strong>
\n<\/strong><\/strong><\/strong><\/strong>[<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/strong>L<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/strong>z<\/span><\/span><\/span><\/span><\/sub><\/span><\/span><\/span><\/strong>,f(<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/strong>R<\/span><\/strong><\/span><\/span><\/span><\/span><\/span><\/span><\/strong>)] = X [P<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/strong>y<\/span><\/span><\/span><\/span><\/span><\/sub><\/span><\/span><\/span><\/strong>,<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/strong><\/strong>f(<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/strong>R<\/span><\/strong><\/span><\/span><\/span><\/span><\/span><\/span><\/strong>)] <\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/strong>\u2013 <\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/strong>Y <\/span><\/strong><\/span><\/span><\/span><\/span><\/span><\/span><\/strong>[<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/strong><\/strong>P<\/span><\/strong><\/span><\/span><\/span><\/span><\/span><\/span><\/strong>x<\/span><\/span><\/span><\/span><\/span><\/sub><\/span><\/span><\/span><\/strong>, f(<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/strong>R) <\/span><\/strong><\/span><\/span><\/span><\/span><\/span><\/span><\/strong>]<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/strong><\/strong><\/strong> ;\u00a0 \u00a0 \u00a0 \u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 (10e)
\n<\/strong><\/strong><\/strong><\/strong><\/strong>[<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/strong>L<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/strong>z<\/span><\/span><\/span><\/span><\/sub><\/span><\/span><\/span><\/strong>,f(<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/strong>R<\/span><\/strong><\/span><\/span><\/span><\/span><\/span><\/span><\/strong>)] = <\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/strong>\u2013 i<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/strong> \u0127 X Y<\/strong><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/strong> \/ R <\/span><\/strong><\/span><\/span><\/span><\/span><\/span><\/span><\/strong>d<\/span><\/span><\/span><\/span><\/span><\/span><\/span>f<\/span><\/strong><\/span><\/span><\/span><\/span><\/span><\/span><\/strong>(<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/strong>R<\/span><\/strong><\/span><\/span><\/span><\/span><\/span><\/span><\/strong>)<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/strong>\/<\/span><\/strong><\/span><\/span><\/span><\/span><\/span><\/span><\/strong>d<\/span><\/span><\/span><\/span><\/span><\/span><\/span>R<\/span><\/strong><\/span><\/span><\/span><\/span><\/span><\/span><\/strong> + i<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/strong><\/strong>\u0127 <\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/strong>Y <\/span><\/strong><\/span><\/span><\/span><\/span><\/span><\/span><\/strong>X \/ R <\/span><\/strong><\/span><\/span><\/span><\/span><\/span><\/span><\/strong>d<\/span><\/span><\/span><\/span><\/span><\/span><\/span>f<\/span><\/strong><\/span><\/span><\/span><\/span><\/span><\/span><\/strong>(<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/strong>R<\/span><\/strong><\/span><\/span><\/span><\/span><\/span><\/span><\/strong>)<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/strong>\/<\/span><\/strong><\/span><\/span><\/span><\/span><\/span><\/span><\/strong>d<\/span><\/span><\/span><\/span><\/span><\/span><\/span>R<\/span><\/strong><\/span><\/span><\/span><\/span><\/span><\/span><\/strong><\/strong><\/strong> ;\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 (10f)
\n<\/strong><\/strong><\/strong><\/strong><\/strong>[<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/strong>L<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/strong>z<\/span><\/span><\/span><\/span><\/sub><\/span><\/span><\/span><\/strong>,f(<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/strong>R<\/span><\/strong><\/span><\/span><\/span><\/span><\/span><\/span><\/strong>)] = 0<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/strong><\/strong><\/strong> ;\u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 \u00a0 \u00a0 \u00a0 \u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 (10g)<\/p>\n

It is left as an exercise for the reader to show that
\n<\/strong><\/strong><\/strong><\/strong><\/strong>[<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/strong><\/strong>L<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/strong>x<\/span><\/span><\/span><\/span><\/sub><\/span><\/span><\/span><\/strong><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/strong>,f(<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/strong>R<\/span><\/strong><\/span><\/span><\/span><\/span><\/span><\/span><\/strong>)] = 0<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/strong><\/strong><\/strong><\/strong><\/strong> ;\u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 \u00a0 \u00a0 \u00a0 \u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 (10h)
\n<\/strong><\/strong><\/strong><\/strong><\/strong><\/strong>and<\/strong>
\n<\/strong><\/strong><\/strong><\/strong><\/strong><\/strong>[<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/strong><\/strong>L<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/strong>y<\/span><\/span><\/span><\/span><\/sub><\/span><\/span><\/span><\/strong><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/strong>,f(<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/strong>R<\/span><\/strong><\/span><\/span><\/span><\/span><\/span><\/span><\/strong>)] = 0<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/strong><\/strong><\/strong><\/strong> ;\u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 \u00a0 \u00a0 \u00a0 \u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 (10i)
\n<\/strong><\/strong><\/strong><\/strong><\/strong><\/strong>It follows that
\n<\/strong><\/strong><\/strong><\/strong><\/strong><\/strong>[<\/span><\/span><\/span><\/span><\/span><\/span>f(<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/strong>R<\/span><\/strong><\/span><\/span><\/span><\/span><\/span><\/span><\/strong>)<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/strong><\/strong>, <\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/strong>L<\/span><\/span><\/span><\/span><\/span><\/span><\/strong><\/strong>2<\/span><\/span><\/span><\/span><\/sup><\/span><\/span><\/strong>] =0.<\/span><\/span><\/span><\/span><\/span><\/span><\/strong><\/strong><\/strong><\/strong> \u00a0\u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 \u00a0 \u00a0 \u00a0 \u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 (10j)
\n<\/strong><\/strong><\/strong><\/strong><\/strong><\/strong>Finally, we show that P<\/span><\/strong><\/span><\/span><\/span><\/span><\/span><\/span><\/strong><\/strong>2<\/span><\/strong><\/span><\/span><\/span><\/sup><\/span><\/span><\/span><\/strong> commutes with every component of angular momentum
\n<\/strong><\/strong><\/strong><\/strong><\/strong><\/strong>[<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/strong>L<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/strong>z<\/span><\/span><\/span><\/span><\/sub><\/span><\/span><\/span><\/strong>, <\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/strong>P<\/span><\/strong><\/span><\/span><\/span><\/span><\/span><\/span><\/strong>x<\/span><\/strong><\/span><\/span><\/span><\/sub><\/span><\/span><\/span><\/strong>2<\/span><\/strong><\/span><\/span><\/span><\/sup><\/span><\/span><\/span><\/strong>] = 2 [<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/strong>X,<\/span><\/strong><\/span><\/span><\/span><\/span><\/span><\/span><\/strong>P<\/span><\/strong><\/span><\/span><\/span><\/span><\/span><\/span><\/strong>x<\/span><\/strong><\/span><\/span><\/span><\/sub><\/span><\/span><\/span><\/strong>]<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/strong><\/strong>P<\/span><\/strong><\/span><\/span><\/span><\/span><\/span><\/span><\/strong>x <\/span><\/strong><\/span><\/span><\/span><\/sub><\/span><\/span><\/span><\/strong>P<\/span><\/strong><\/span><\/span><\/span><\/span><\/span><\/span><\/strong>y<\/span><\/span><\/span><\/span><\/span><\/sub><\/span><\/span><\/span><\/strong> \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 \u00a0 \u00a0 \u00a0 \u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 (11a)
\n<\/strong><\/strong><\/strong><\/strong><\/strong>[<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/strong>L<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/strong>z<\/span><\/span><\/span><\/span><\/sub><\/span><\/span><\/span><\/strong>, <\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/strong>P<\/span><\/strong><\/span><\/span><\/span><\/span><\/span><\/span><\/strong>z<\/span><\/strong><\/span><\/span><\/span><\/sub><\/span><\/span><\/span><\/strong>] = 0<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/strong> \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 \u00a0 \u00a0 \u00a0 \u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 \u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 (11b)
\n<\/strong><\/strong><\/strong><\/strong><\/strong>[<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/strong>L<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/strong>z<\/span><\/span><\/span><\/span><\/sub><\/span><\/span><\/span><\/strong>, <\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/strong>P<\/span><\/strong><\/span><\/span><\/span><\/span><\/span><\/span><\/strong>z<\/span><\/strong><\/span><\/span><\/span><\/sub><\/span><\/span><\/span><\/strong>2<\/span><\/strong><\/span><\/span><\/span><\/sup><\/span><\/span><\/span><\/strong>] = 0<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/strong> \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 \u00a0 \u00a0 \u00a0 \u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 (11c)
\n<\/strong><\/strong><\/strong><\/strong><\/strong>[<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/strong>L<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/strong>z<\/span><\/span><\/span><\/span><\/sub><\/span><\/span><\/span><\/strong>, <\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/strong>P<\/span><\/strong><\/span><\/span><\/span><\/span><\/span><\/span><\/strong>x<\/span><\/strong><\/span><\/span><\/span><\/sub><\/span><\/span><\/span><\/strong>2<\/span><\/strong><\/span><\/span><\/span><\/sup><\/span><\/span><\/span><\/strong>] = 2 i\u0127 <\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/strong>P<\/span><\/strong><\/span><\/span><\/span><\/span><\/span><\/span><\/strong>x <\/span><\/strong><\/span><\/span><\/span><\/sub><\/span><\/span><\/span><\/strong>P<\/span><\/strong><\/span><\/span><\/span><\/span><\/span><\/span><\/strong>y<\/span><\/span><\/span><\/span><\/span><\/sub><\/span><\/span><\/span><\/strong> \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 \u00a0 \u00a0 \u00a0 \u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 \u00a0\u00a0\u00a0\u00a0 (11d)
\n<\/strong><\/strong><\/strong><\/strong><\/strong>[<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/strong>L<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/strong>z<\/span><\/span><\/span><\/span><\/sub><\/span><\/span><\/span><\/strong>, <\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/strong>P<\/span><\/strong><\/span><\/span><\/span><\/span><\/span><\/span><\/strong>y<\/span><\/strong><\/span><\/span><\/span><\/sub><\/span><\/span><\/span><\/strong>2<\/span><\/strong><\/span><\/span><\/span><\/sup><\/span><\/span><\/span><\/strong>] = [<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/strong>X<\/span><\/strong><\/span><\/span><\/span><\/span><\/span><\/span><\/strong><\/strong>P<\/span><\/strong><\/span><\/span><\/span><\/span><\/span><\/span><\/strong>y<\/span><\/span><\/span><\/span><\/span><\/sub><\/span><\/span><\/span><\/strong>,<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/strong>P<\/span><\/strong><\/span><\/span><\/span><\/span><\/span><\/span><\/strong>y<\/span><\/span><\/span><\/span><\/span><\/sub><\/span><\/span><\/span><\/strong>2<\/span><\/span><\/span><\/span><\/span><\/sup><\/span><\/span><\/span><\/strong>]<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/strong> \u2013 [<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/strong>Y,<\/span><\/strong><\/span><\/span><\/span><\/span><\/span><\/span><\/strong>P<\/span><\/strong><\/span><\/span><\/span><\/span><\/span><\/span><\/strong>y<\/span><\/span><\/span><\/span><\/span><\/sub><\/span><\/span><\/span><\/strong>2<\/span><\/span><\/span><\/span><\/span><\/sup><\/span><\/span><\/span><\/strong>]<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/strong><\/strong>P<\/span><\/strong><\/span><\/span><\/span><\/span><\/span><\/span><\/strong>x<\/span><\/span><\/span><\/span><\/span><\/sub><\/span><\/span><\/span><\/strong> \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 \u00a0 \u00a0 \u00a0 \u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 (11e)
\n<\/strong><\/strong><\/strong><\/strong><\/strong><\/strong>[<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/strong>L<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/strong>z<\/span><\/span><\/span><\/span><\/sub><\/span><\/span><\/span><\/strong>, <\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/strong>P<\/span><\/strong><\/span><\/span><\/span><\/span><\/span><\/span><\/strong>y<\/span><\/strong><\/span><\/span><\/span><\/sub><\/span><\/span><\/span><\/strong>2<\/span><\/strong><\/span><\/span><\/span><\/sup><\/span><\/span><\/span><\/strong>] = 0 -2 i\u0127 <\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/strong><\/strong>P<\/span><\/strong><\/span><\/span><\/span><\/span><\/span><\/span><\/strong>y <\/span><\/span><\/span><\/span><\/span><\/sub><\/span><\/span><\/span><\/strong><\/strong>P<\/span><\/strong><\/span><\/span><\/span><\/span><\/span><\/span><\/strong>x<\/span><\/strong><\/span><\/span><\/span><\/sub><\/span><\/span><\/span><\/strong> \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 \u00a0 \u00a0 \u00a0 \u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 (11f)
\n<\/strong><\/strong><\/strong><\/strong><\/strong>[<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/strong>L<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/strong>z<\/span><\/span><\/span><\/span><\/sub><\/span><\/span><\/span><\/strong>, <\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/strong>P<\/span><\/strong><\/span><\/span><\/span><\/span><\/span><\/span><\/strong>x<\/span><\/strong><\/span><\/span><\/span><\/sub><\/span><\/span><\/span><\/strong>2 <\/span><\/strong><\/span><\/span><\/span><\/sup><\/span><\/span><\/span><\/strong>+ P<\/span><\/strong><\/span><\/span><\/span><\/span><\/span><\/span><\/strong>y<\/span><\/strong><\/span><\/span><\/span><\/sub><\/span><\/span><\/span><\/strong>2<\/span><\/strong><\/span><\/span><\/span><\/sup><\/span><\/span><\/span><\/strong>] = 2 i\u0127 [<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/strong>P<\/span><\/strong><\/span><\/span><\/span><\/span><\/span><\/span><\/strong>x <\/span><\/strong><\/span><\/span><\/span><\/sub><\/span><\/span><\/span><\/strong>,<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/strong>P<\/span><\/strong><\/span><\/span><\/span><\/span><\/span><\/span><\/strong>y<\/span><\/span><\/span><\/span><\/span><\/sub><\/span><\/span><\/span><\/strong>]<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/strong>.<\/strong> \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 \u00a0 \u00a0 \u00a0 \u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 (11g)
\n<\/strong><\/strong><\/strong><\/strong><\/strong>It follows that<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/strong>
\n<\/strong><\/strong><\/strong><\/strong><\/strong>[<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/strong>L<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/strong>z<\/span><\/span><\/span><\/span><\/sub><\/span><\/span><\/span><\/strong>, <\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/strong>P<\/span><\/strong><\/span><\/span><\/span><\/span><\/span><\/span><\/strong><\/strong>2<\/span><\/strong><\/span><\/span><\/span><\/sup><\/span><\/span><\/span><\/strong>] = 0.<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/strong> \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 \u00a0 \u00a0 \u00a0 \u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 \u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 (11h)
\n\u00a0<\/strong><\/strong><\/strong><\/strong><\/strong><\/strong>It is left as an exercise for the reader t show that<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/strong>
\n<\/strong><\/strong><\/strong><\/strong><\/strong>[<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/strong>L<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/strong>x<\/span><\/span><\/span><\/span><\/sub><\/span><\/span><\/span><\/strong>, <\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/strong>P<\/span><\/strong><\/span><\/span><\/span><\/span><\/span><\/span><\/strong><\/strong>2<\/span><\/strong><\/span><\/span><\/span><\/sup><\/span><\/span><\/span><\/strong>] = 0 <\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/strong> \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 \u00a0 \u00a0 \u00a0 \u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 \u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 (11j) <\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/strong>
\n<\/strong><\/strong><\/strong><\/strong><\/strong><\/strong><\/strong><\/strong><\/strong><\/strong><\/strong><\/strong>and <\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/strong><\/strong>
\n<\/strong><\/strong><\/strong><\/strong><\/strong><\/strong>[<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/strong>L<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/strong>y<\/span><\/span><\/span><\/span><\/sub><\/span><\/span><\/span><\/strong>, <\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/strong>P<\/span><\/strong><\/span><\/span><\/span><\/span><\/span><\/span><\/strong><\/strong>2<\/span><\/strong><\/span><\/span><\/span><\/sup><\/span><\/span><\/span><\/strong>] = 0.<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/strong> \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 \u00a0 \u00a0 \u00a0 \u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 \u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 (11k)<\/p>\n

Using the form of the Hamiltonian (1b) it follows that
\n<\/strong><\/strong>[L<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/strong>x<\/span><\/span><\/span><\/span><\/sub><\/span><\/span><\/span><\/strong>, H<\/strong>] = 0 ;\u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 \u00a0 \u00a0 \u00a0 \u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 \u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 (12a)
\n<\/strong> <\/strong>[L<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/strong>y<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/strong><\/strong>, H<\/strong>] = 0 ;\u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 \u00a0 \u00a0 \u00a0 \u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 \u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 (12b)
\n<\/strong>[L<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/strong>z<\/span><\/span><\/span><\/span><\/sub><\/span><\/span><\/span><\/strong>, H<\/strong>] = 0 ;\u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 \u00a0 \u00a0 \u00a0 \u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 \u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 (12c)
\nand therefore
\n<\/strong>[L<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/strong><\/strong><\/strong><\/strong>2<\/span><\/strong><\/span><\/span><\/span><\/sup><\/span><\/span><\/span><\/strong><\/strong><\/strong><\/strong>, H<\/strong>] = 0.\u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 \u00a0 \u00a0 \u00a0 \u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 \u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 (12d)<\/p>\n

The Casimir operators for the hydrogen atom are therefore\u00a0 H<\/strong>, L<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/strong><\/strong><\/strong><\/strong>2<\/span><\/strong><\/span><\/span><\/span><\/sup><\/span><\/span><\/span><\/strong>, and L<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/strong>z<\/span><\/span><\/span><\/span><\/sub><\/span><\/span><\/span><\/strong> . The choice of the latter component of angular momentum is arbitrary, but by convention it is taken as L<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/strong>z<\/span><\/span><\/span><\/span><\/sub><\/span><\/span><\/span><\/strong> .<\/p>\n

In the next blog in this series on the hydrogen atom, we will derive solutions for three simultaneous eigenvalues of the Casimir operators:
\n<\/strong>H<\/strong>|\u03a6> =E<\/span><\/span>n<\/span><\/span><\/sub> |\u03a6>, \u00a0 \u00a0 \u00a0 \u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 \u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 (12e)
\n<\/strong>L<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/strong><\/strong><\/strong><\/strong>2<\/span><\/strong><\/span><\/span><\/span><\/sup><\/span><\/span><\/span><\/strong><\/strong><\/strong><\/strong>|\u03a6> =\u03b1\u00a0<\/span><\/span><\/sub> |\u03a6>
\n<\/strong><\/strong><\/strong><\/strong>L<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/strong>z<\/span><\/span><\/span><\/span><\/sub><\/span><\/span><\/span><\/strong>|\u03a6> =\u03b2\u00a0<\/span><\/span><\/sub> |\u03a6>.\u00a0 \u00a0 \u00a0 \u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 \u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 (12g)
\n<\/strong><\/strong><\/p>\n

 <\/p>\n

 <\/p>\n

 <\/p>\n

 <\/p>\n

 <\/p>\n","protected":false},"excerpt":{"rendered":"

\u00a9Fernando Caracena 2013 On a previous blog on the one-dimensional harmonic oscillator, we showed that the algebra of non-commuting operators in quantum mechanics could be exploited systematically to solve for the entire spectrum of state vectors and energy eigenvalues. Algebra … Continue reading →<\/span><\/a><\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[31,10,72,73],"tags":[],"class_list":["post-1043","post","type-post","status-publish","format-standard","hentry","category-algebra","category-mathematics","category-quantum-physics","category-state-vectors"],"_links":{"self":[{"href":"http:\/\/www.ghyzmo.com\/wp-json\/wp\/v2\/posts\/1043","targetHints":{"allow":["GET"]}}],"collection":[{"href":"http:\/\/www.ghyzmo.com\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"http:\/\/www.ghyzmo.com\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"http:\/\/www.ghyzmo.com\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"http:\/\/www.ghyzmo.com\/wp-json\/wp\/v2\/comments?post=1043"}],"version-history":[{"count":10,"href":"http:\/\/www.ghyzmo.com\/wp-json\/wp\/v2\/posts\/1043\/revisions"}],"predecessor-version":[{"id":1067,"href":"http:\/\/www.ghyzmo.com\/wp-json\/wp\/v2\/posts\/1043\/revisions\/1067"}],"wp:attachment":[{"href":"http:\/\/www.ghyzmo.com\/wp-json\/wp\/v2\/media?parent=1043"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"http:\/\/www.ghyzmo.com\/wp-json\/wp\/v2\/categories?post=1043"},{"taxonomy":"post_tag","embeddable":true,"href":"http:\/\/www.ghyzmo.com\/wp-json\/wp\/v2\/tags?post=1043"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}