Spherical Coordinates: Difference between revisions

	Quantum Mechanics A
	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.
	Physical Basis of Quantum Mechanics
	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
	Schrödinger Equation
	Brief Derivation of Schrödinger Equation; Relation Between the Wave Function and Probability Density; Stationary States; Heisenberg Uncertainty Principle; Some Consequences of the Uncertainty Principle
	Operators, Eigenfunctions, and Symmetry
	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
	Motion in One Dimension
	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
	Discrete Eigenvalues and Bound States
	Harmonic Oscillator Spectrum and Eigenstates; Analytical Method for Solving the Simple Harmonic Oscillator; Coherent States; Charged Particles in an Electromagnetic Field; WKB Approximation
	Time Evolution and the Pictures of Quantum Mechanics
	The Heisenberg Picture: Equations of Motion for Operators; The Interaction Picture; The Virial Theorem
	Angular Momentum
	Commutation Relations; Angular Momentum as a Generator of Rotations in 3D; Spherical Coordinates; Eigenvalue Quantization; Orbital Angular Momentum Eigenfunctions
	Central Forces
	General Formalism; Free Particle in Spherical Coordinates; Spherical Well; Isotropic Harmonic Oscillator; Hydrogen Atom; WKB in Spherical Coordinates
	The Path Integral Formulation of Quantum Mechanics
	Feynman Path Integrals; The Free-Particle Propagator; Propagator for the Harmonic Oscillator
	Continuous Eigenvalues and Collision Theory
	Differential Cross Section and the Green's Function Formulation of Scattering; Central Potential Scattering and Phase Shifts; Coulomb Potential Scattering

Revision as of 22:25, 28 August 2013

We now write down the Cartesian components of the angular momentum operator in spherical coordinates. We will make use of this result later in determining the eigenfunctions of the angular momentum squared and of one of its components.

The Cartesian coordinates 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 x,\!} 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 y,\!} 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 z\!} can be written in terms of the spherical coordinates 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 r,\!} 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 \theta,\!} 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 \phi\!} as follows:

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 x=r\sin\theta\cos\phi,\!} 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 y=r\sin\theta\sin\phi,\!} 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 z=r\cos\theta\!}

Let us start with the 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 x\!} component of the angular momentum, 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}_x.} In Cartesian coordinates, this is

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}_x=-i\hbar\left (y\frac{\partial}{\partial z}-z\frac{\partial}{\partial y}\right ).}

If we make use of the chain rule, then we obtain

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}_{x} = -i\hbar\left( -\sin\phi \frac{\partial}{\partial \theta} \, -\cot\theta\cos\phi \frac{\partial}{\partial\phi} \! \right).}

Similarly, the 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 y\!} 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 z\!} components may be found to be

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}_{y} = -i\hbar\left(\cos\phi \frac{\partial}{\partial\theta} - \cot\theta\sin\phi \frac{\partial}{\partial\phi} \! \right) }

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} = -i\hbar\frac{\partial}{\partial \phi}.}

Problem

(Richard L. Liboff, Introductory Quantum Mechanics, 2nd Edition, pp. 377-379)

Show, using the above results, that the operator,

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{R}_{\Delta\phi}=\exp \left (\frac{i}{\hbar}\Delta\phi\hat{L}_z\right ),}

when applied to a function 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 f(\phi)\!} of the azimuthal angle 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 \phi,\!} rotates the angle 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 \phi\!} to 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 \phi+\Delta\phi.} That is, 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{R}_{\Delta\phi}f(\phi)=f(\phi+\Delta\phi).}

Solution

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 We now write down the Cartesian components of the angular momentum operator in spherical coordinates.  We will make use of this result later in determining the eigenfunctions of the angular momentum squared and of one of its components.
-The Cartesian coordinates <math>x,\!</math <math>y,\!</math> and <math>z\!</math> can be written in terms of the spherical coordinates <math>r,\!</math> <math>\theta,\!</math> and <math>\phi\!</math> as follows:
+The Cartesian coordinates <math>x,\!</math> <math>y,\!</math> and <math>z\!</math> can be written in terms of the spherical coordinates <math>r,\!</math> <math>\theta,\!</math> and <math>\phi\!</math> as follows:
 <math>x=r\sin\theta\cos\phi,\!</math> <math>y=r\sin\theta\sin\phi,\!</math> <math>z=r\cos\theta\!</math>
-Denote the state <math> \langle \mathbf{r} \!\, | = \langle \mathbf{r}\! \,  \theta \phi |  </math>
+Let us start with the <math>x\!</math> component of the angular momentum, <math>\hat{L}_x.</math>  In Cartesian coordinates, this is
-If you choose <math> \alpha \! </math> along the z-axis then the only coordinate that will change is phi such that <math> \phi \rightarrow \phi + \alpha </math>.  Now the state is written as:
+<math>\hat{L}_x=-i\hbar\left (y\frac{\partial}{\partial z}-z\frac{\partial}{\partial y}\right ).</math>
-<math> \langle \mathbf{r}\! \, \theta \phi | \left(1 + \frac{i}{\hbar} \alpha L_{z}\right) = \langle \mathbf{r} \! \, \theta \phi + \alpha | </math>
+If we make use of the chain rule, then we obtain
-Working to first order in alpha the right hand side becomes:
+<math>\hat{L}_{x} = -i\hbar\left( -\sin\phi \frac{\partial}{\partial \theta} \, -\cot\theta\cos\phi \frac{\partial}{\partial\phi} \! \right).</math>
-<math> \langle \mathbf{r}\! \, \theta \phi | + \alpha \frac{\partial}{\partial \phi} \langle \mathbf{r}\! \, \theta \phi | </math>
+Similarly, the <math>y\!</math> and <math>z\!</math> components may be found to be
-Therefore <math> L_{z} = \frac{\hbar}{i} \frac{\partial}{\partial \phi} </math>
+<math>\hat{L}_{y} = -i\hbar\left(\cos\phi \frac{\partial}{\partial\theta} - \cot\theta\sin\phi \frac{\partial}{\partial\phi} \! \right) </math>
+and
-Now choose <math> \alpha \! </math> along the x-axis then the cartesian coordinates are changed such that:
+<math>\hat{L}_{z} = -i\hbar\frac{\partial}{\partial \phi}.</math>
-<math> x \rightarrow x  \!  </math> ,
-<math> y \rightarrow y - \alpha z  \!</math>, and
-<math> z \rightarrow z + \alpha y \! </math>,
-from these transformations it can be determined that <math> \delta \theta = -\alpha\sin\phi \! </math> since <math> \delta z = \alpha y \! </math>
-and since x does not change it can be determined that <math> \delta \phi  = -\alpha\cot\theta \cos\phi \! </math>.
-This means that the original state is now written as:
-<math> \langle \mathbf{r}\! \, \theta \phi | \left(1 + \frac{i}{\hbar} \alpha L_{x}\right) = \langle \mathbf{r} \! \, \,\theta - \alpha\sin\phi \,  \phi - \alpha\cot\theta\cos\phi | </math>
-Expanding the right hand side of the above equation as before to the first order of alpha the whole equation becomes:
-<math> \langle \mathbf{r}\! \, \theta \phi | L_{x} = \frac{\hbar}{i} \left( -\sin\phi \frac{\partial}{\partial \theta} \, -\cot\theta\cos\phi \frac{\partial}{\partial\phi} \! \right)   \langle \mathbf{r}\! \, \theta \phi | </math>
-Therfore <math> L_{x} = \frac{\hbar}{i}\left( -\sin\phi \frac{\partial}{\partial \theta} \, -\cot\theta\cos\phi \frac{\partial}{\partial\phi} \! \right) </math>
-Using the same techinque, choose <math> \alpha \! </math> along the y-axis and the coordinates will change in a similar fashion such that it can be shown that <math> L_{y} = \frac{\hbar}{i} \left(\cos\phi \frac{\partial}{\partial\theta} - \cot\theta\sin\phi \frac{\partial}{\partial\phi} \! \right) </math>
 ==Problem==

Spherical Coordinates: Difference between revisions

Revision as of 22:25, 28 August 2013

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