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{{Quantum Mechanics A}}
{{Quantum Mechanics A}}
The motivation for exploring eigenvalue quantization comes form wanting to solve the energy eigenvalue problem for a particle in a central potential.  It is not possible, in general, to specify and measure more than one component <math> \hat{\mathbf{n}}\cdot\hat{\mathbf{L}}</math> of the orbital angular momentum.  It is, however, possible to specify <math> \hat{\mathbf{L}}^2 </math> simulataneously with any one component of <math>\hat{\mathbf{L}}.</math>  We typically choose <math>\hat{L}_z.</math>  A central potential yields a Hamiltonian that commutes with <math>\hat{\mathbf{L}},</math> and thus the energy eigenstates of the system may be chosen to be eigenvectors of <math>  \hat{\mathbf{L}}^2 </math> and one component of <math>\hat{\mathbf{L}},</math> usually <math>\hat{L}_z.</math>
The motivation for exploring eigenvalue quantization comes form wanting to solve the energy eigenvalue problem for a particle in a central potential.  It is not possible, in general, to specify and measure more than one component <math> \hat{\mathbf{n}}\cdot\hat{\mathbf{L}}</math> of the orbital angular momentum.  It is, however, possible to specify <math> \hat{\mathbf{L}}^2 </math> simulataneously with any one component of <math>\hat{\mathbf{L}},</math> since <math>\hat{\mathbf{L}}^2</math> commutes with all of its Cartesian components, as we saw earlier. We typically choose <math>\hat{L}_z.</math>  A central potential yields a Hamiltonian that commutes with <math>\hat{\mathbf{L}},</math> and thus the energy eigenstates of the system may be chosen to be eigenvectors of <math>  \hat{\mathbf{L}}^2 </math> and one component of <math>\hat{\mathbf{L}},</math> usually <math>\hat{L}_z.</math>


The quantization of angular momentum follows simply from the above commutation relations. Define <math>\mathbf{L}^2\!</math> by:
The quantization of angular momentum follows simply from the [[Commutation Relations|commutation relations]] derived earlier. Recall that <math>\mathbf{L}^2\!</math> is given by
:<math>\mathbf{L}^2=L_x^2+L_y^2+L_z^2</math>


Since <math>\mathbf{L}^2\!</math> is a scalar, it commutes with each component of angular momentum.
<math>\hat{\mathbf{L}}^2=\hat{L}_x^2+\hat{L}_y^2+\hat{L}_z^2.</math>


Now Define a change of operators as follows:
Let us now define the operators, <math>\hat{L}_\pm=\hat{L}_x\pm i\hat{L}_y.\!</math>
:<math>L_+=L_x+iL_y\!</math>
:<math>L_-=L_x-iL_y\!</math>


The choice of these two operators are choosen because their commutation relations with <math> L_z \!</math> and one another only result in soultions with <math> L_- \!</math>, <math>  L_+ \!</math>, and <math>  L_z \!</math>.
The choice of these two operators are choosen because their commutation relations with <math>\hat{L}_z\!</math> and one another only involve themselves and <math>\hat{L}_z.\!</math>


From the commutation relations we get
We may use the commutation relations derived earlier to show that
:<math>L_+L_-=\mathbf{L}^2-L_z^2+\hbar L_z</math>
<math>\hat{L}_+\hat{L}_-=\hat{\mathbf{L}}^2-\hat{L}_z^2+\hbar\hat{L}_z</math>


Similarly,
and
:<math>L_-L_+=\mathbf{L}^2-L_z^2-\hbar L_z</math>
 
<math>\hat{L}_-\hat{L}_+=\hat{\mathbf{L}}^2-\hat{L}_z^2-\hbar\hat{L}_z.</math>
                                                              
                                                              
Thus,
Therefore,
:<math>[L_+,L_-]=L_+L_--L_-L_+=2\hbar L_z</math>
 
<math>[\hat{L}_+,\hat{L}_-]=\hat{L}_+\hat{L}_--\hat{L}_-\hat{L}_+=2\hbar\hat{L}_z.</math>


And
We may also show that
:<math>[L_z,L_-]=L_zL_--L_-L_z=-\hbar L_-</math>
<math>[\hat{L}_z,\hat{L}_\pm]=\hat{L}_z\hat{L}_\pm-\hat{L}_\pm\hat{L}_z=\pm\hbar\hat{L}_\pm.</math>


Also, It is easy to show that:
We may also easily see that <math>[\hat{\mathbf{L}}^2,\hat{L}_\pm]=0.</math>
:<math>[L^2,L_\pm]=0</math>


Let <math>L'_z\!</math> be an eigenvalue of <math>L_z\!</math>.
Let <math>L'_z\!</math> be an eigenvalue of <math>L_z\!</math>.

Revision as of 22:42, 28 August 2013

Quantum Mechanics A
SchrodEq.png
Schrödinger Equation
The most fundamental equation of quantum mechanics; given a Hamiltonian Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathcal{H}} , it describes how a state Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle |\Psi\rangle} evolves in time.
Basic Concepts and Theory of Motion
UV Catastrophe (Black-Body Radiation)
Photoelectric Effect
Stability of Matter
Double Slit Experiment
Stern-Gerlach Experiment
The Principle of Complementarity
The Correspondence Principle
The Philosophy of Quantum Theory
Brief Derivation of Schrödinger Equation
Relation Between the Wave Function and Probability Density
Stationary States
Heisenberg Uncertainty Principle
Some Consequences of the Uncertainty Principle
Linear Vector Spaces and Operators
Commutation Relations and Simultaneous Eigenvalues
The Schrödinger Equation in Dirac Notation
Transformations of Operators and Symmetry
Time Evolution of Expectation Values and Ehrenfest's Theorem
One-Dimensional Bound States
Oscillation Theorem
The Dirac Delta Function Potential
Scattering States, Transmission and Reflection
Motion in a Periodic Potential
Summary of One-Dimensional Systems
Harmonic Oscillator Spectrum and Eigenstates
Analytical Method for Solving the Simple Harmonic Oscillator
Coherent States
Charged Particles in an Electromagnetic Field
WKB Approximation
The Heisenberg Picture: Equations of Motion for Operators
The Interaction Picture
The Virial Theorem
Commutation Relations
Angular Momentum as a Generator of Rotations in 3D
Spherical Coordinates
Eigenvalue Quantization
Orbital Angular Momentum Eigenfunctions
General Formalism
Free Particle in Spherical Coordinates
Spherical Well
Isotropic Harmonic Oscillator
Hydrogen Atom
WKB in Spherical Coordinates
Feynman Path Integrals
The Free-Particle Propagator
Propagator for the Harmonic Oscillator
Differential Cross Section and the Green's Function Formulation of Scattering
Central Potential Scattering and Phase Shifts
Coulomb Potential Scattering

The motivation for exploring eigenvalue quantization comes form wanting to solve the energy eigenvalue problem for a particle in a central potential. It is not possible, in general, to specify and measure more than one component Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \hat{\mathbf{n}}\cdot\hat{\mathbf{L}}} of the orbital angular momentum. It is, however, possible to specify Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \hat{\mathbf{L}}^2 } simulataneously with any one component of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \hat{\mathbf{L}},} since Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \hat{\mathbf{L}}^2} commutes with all of its Cartesian components, as we saw earlier. We typically choose Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \hat{L}_z.} A central potential yields a Hamiltonian that commutes with Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \hat{\mathbf{L}},} and thus the energy eigenstates of the system may be chosen to be eigenvectors of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \hat{\mathbf{L}}^2 } and one component of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \hat{\mathbf{L}},} usually Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \hat{L}_z.}

The quantization of angular momentum follows simply from the commutation relations derived earlier. Recall that Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf{L}^2\!} is given by

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \hat{\mathbf{L}}^2=\hat{L}_x^2+\hat{L}_y^2+\hat{L}_z^2.}

Let us now define the operators, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \hat{L}_\pm=\hat{L}_x\pm i\hat{L}_y.\!}

The choice of these two operators are choosen because their commutation relations with Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \hat{L}_z\!} and one another only involve themselves and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \hat{L}_z.\!}

We may use the commutation relations derived earlier to show that Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \hat{L}_+\hat{L}_-=\hat{\mathbf{L}}^2-\hat{L}_z^2+\hbar\hat{L}_z}

and

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \hat{L}_-\hat{L}_+=\hat{\mathbf{L}}^2-\hat{L}_z^2-\hbar\hat{L}_z.}

Therefore,

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle [\hat{L}_+,\hat{L}_-]=\hat{L}_+\hat{L}_--\hat{L}_-\hat{L}_+=2\hbar\hat{L}_z.}

We may also show that Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle [\hat{L}_z,\hat{L}_\pm]=\hat{L}_z\hat{L}_\pm-\hat{L}_\pm\hat{L}_z=\pm\hbar\hat{L}_\pm.}

We may also easily see that Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle [\hat{\mathbf{L}}^2,\hat{L}_\pm]=0.}

Let Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle L'_z\!} be an eigenvalue of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle L_z\!} .

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \langle L_z'|L_+L_-|L_z'\rangle=(\beta-L_z'^2+\hbar L_z')\langle L_z'|L_z'\rangle}

Since the left hand side of the above equation is the square of the length of a ket, it has to be non-negative. Therefore

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf{L}^2-L_z'^2+\hbar L_z'\ge0}
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \Rightarrow \beta+\frac{\hbar^2}{4}\ge (L_z'-\frac{\hbar}{2})^2}
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \Rightarrow \beta+\frac{\hbar^2}{4}\ge 0}

Defining the number Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle k\!} by

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle k+\frac{\hbar}{2}=\sqrt{\beta+\frac{\hbar^2}{4}}}
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \Rightarrow k\ge -\frac{\hbar}{2}}

The inequality 5.1.13 becomes

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle k+\frac{\hbar}{2}\ge |L_z'-\frac{\hbar}{2}|}
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \Rightarrow k+\hbar\ge L_z'\ge -k}

Similarly, from equation 5.1.10, we get

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \langle L_z'|L_-L_+|L_z'\rangle=(\beta-L_z'^2-\hbar L_z')\langle L_z'|L_z'\rangle}
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \Rightarrow \mathbf{L}^2-L_z'^2-\hbar L_z'\ge0}
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \Rightarrow k\ge L_z'\ge -k-\hbar}

This result, combined with 5.1.15 shows that

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle k\ge 0}

and

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle k\ge L_z'\ge -k}

From 5.1.12

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle L_z L_-|L_z'\rangle=(L_- L_z-\hbar L_-)|L_z'\rangle=(L_z'-\hbar)L_-|L_z'\rangle}

Now, if Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle L_z'\ne 0} , then Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle L_-|L_z'\rangle} is an eigenket of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle L_z\!} belonging to the eigenvalue Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle L_z'-\hbar} . Similarly, if Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle L_z'-\hbar\ne -k} , then Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle L_z'-2\hbar} is another eigenvalue of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle L_z\!} , and so on. In this way, we get a series of eigenvalues which must terminate from 5.1.16, and can terminate only with the value Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \!-k} . Similarly, using the complex conjugate of 5.1.12, we get that Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle L_z',L_z'+\hbar,L_z'+2\hbar,...\!} are eigenvalues of L'z. Thus we may conclude that Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 2k\!} is an integral multiple of the Planck's constant, and that the eigenvalues are:

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle k, k-\hbar,k-2\hbar,...,-k+\hbar,-k}

If Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle |m\rangle\!} is an eigenstate of with eigenvalue , then

Which means that or raises or lowers the component of the angular momentum by .

Calculation of two angular momentum expressions

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