Transformations of Operators and Symmetry

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Quantum Mechanics A
Schrödinger Equation
The most fundamental equation of quantum mechanics; given a Hamiltonian \mathcal{H}, it describes how a state |\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

Transformations of Operators

In the previous section, we discussed operators as transformations of vectors. In many cases, however, we will be interested in how operators, observables in particular, will transform under the action of another operator. Given an operator \hat{A} and a transformation \hat{T}, we define the transformed operator \hat{A}' as follows. Given the relation,

\hat{A}|\psi\rangle=|\phi\rangle,

between two vectors |\psi\rangle and |\phi\rangle, the operator \hat{A}' is the operator giving the relation between |\psi'\rangle=\hat{T}|\psi\rangle and |\phi'\rangle=\hat{U}|\phi\rangle; i.e.,

\hat{A}'|\psi'\rangle=|\phi'\rangle.

To find \hat{A}', let us first act on both sides of the original relation with \hat{T}:

\hat{T}\hat{A}|\psi\rangle=\hat{T}|\phi\rangle

We now introduce the identity between \hat{A} and |\psi\rangle in the form, \hat{T}^{-1}\hat{T}:

\hat{T}\hat{A}\hat{T}^{-1}\hat{T}|\psi\rangle=\hat{T}|\phi\rangle

Using the above definitions of |\psi'\rangle and |\phi'\rangle, we may write this as

\hat{T}\hat{A}\hat{T}^{-1}|\psi'\rangle=|\phi'\rangle

We see then that the transformed operator \hat{A}'=\hat{T}\hat{A}\hat{T}^{-1}. In matrix form, this would simply correspond to a similarity transformation of \hat{A}.

Of particular importance is the case in which \hat{T} is unitary and \hat{A} is an observable. This is because, in addition to preserving the normalization of the state vectors, as mentioned in the previous section, unitary transformations also preserve the Hermitian nature of \hat{A}:

\hat{A}'^{\dagger}=(\hat{T}\hat{A}\hat{T}^\dagger)^\dagger=\hat{T}\hat{A}^\dagger\hat{T}^\dagger=\hat{T}\hat{A}\hat{T}^\dagger=\hat{A}'

Symmetry and its Role in Quantum Mechanics

Having discussed the transformation of operators, we will now apply our results to discuss symmetries of the Hamiltonian, a very important topic. As alluded to in the previous section, identifying the symmetries of the Hamiltonian will allow us to greatly simplify the problem at hand. In addition, in both classical and quantum mechanics, symmetry transformations become important due to their relation to conserved quantities via Noether's Theorem. In quantum mechanics, the importance of symmetries is further enhanced by the fact that measurements of conserved quantities can be exact in spite of the probabilistic nature of quantum predictions.

Given a unitary transformation \hat{U}, we say that it is a symmetry of the Hamiltonian if it leaves the Hamiltonian invariant; i.e., if \hat{H}'=\hat{U}\hat{H}\hat{U}^\dagger=\hat{H}. We will now show that, if a transformation is a symmetry of the Hamiltonian, then it commutes with the Hamiltonian. To see this, let us take the relation,

\hat{H}|\psi\rangle=|\phi\rangle,

and act on both sides with \hat{U}:

\hat{U}\hat{H}|\psi\rangle=\hat{U}|\phi\rangle

Now, if \hat{U} is a symmetry of the Hamiltonian, then it must also be true that

\hat{H}\hat{U}|\psi\rangle=\hat{U}|\phi\rangle.

Subtracting these two equations, we see that, because |\psi\rangle is arbitrary, the Hamiltonian commutes with the transformation operator; i.e., [\hat{H},\hat{U}]=0.

This is a very important result; we know that, if two operators commute, then it is possible to simultaneously diagonalize them. This implies that every symmetry of the Hamiltonian has a "good quantum number" associated with it that we may use to describe the eigenstates of the Hamiltonian.

To help illustrate this fact, let us consider the parity, or inversion, operator, \hat{P}:

\hat{P}f(x)=f(-x).

The parity operator commutes with the Hamiltonian \hat{H} if the potential is symmetric; i.e., \hat{V}(x)=\hat{V}(-x). Since the two commute, the eigenfunctions of the Hamiltonian can be chosen to be eigenfunctions of the parity operator. This means that, if the potential is symmetric, then the eigenstates of the Hamiltonian can be chosen to have definite parity (even or odd).

Problem

(From a Quantum Mechanics assignment in the Department of Physics, UF)

Consider an N state system, with N even and the states labeled as |1\rangle, |2\rangle, \ldots, |N\rangle, described by the Hamiltonian,

\hat{H}=\sum_{n=1}^{N} (|n\rangle \langle n+1| + |n+1\rangle \langle n|).

Notice that the Hamiltonian, in this form, is manifestly Hermitian. Assume periodic boundary conditions; i.e, |N+1\rangle = |1\rangle. One may therefore think of this Hamiltonian as describing a particle on a circle.

(a) Define the translation operator, \hat{T}, as taking |1\rangle \to |2\rangle, |2\rangle \to |3\rangle ,...,|N\rangle \to |1\rangle. Write \hat{T} in a form like \hat{H} in the first equation and show that \hat{T} is both unitary and commutes with \hat{H}, thus showing that \hat{T} is a symmetry of the Hamiltonian.

(b) Find the eigenstates of \hat{T} by using wavefunctions of the form,

|\psi(k)\rangle = \sum_{n=1}^{N} e^{ikn}|n\rangle.

What are the eigenvalues associated with these eigenstates? Do all these eigenstates have to be eigenstates of \hat{H} as well? If not, do any of these eigenstates have to be eigenstates of \hat{H}? Explain your answer.

(c) Next, consider the operator \hat{F} which takes |n\rangle \to |N+1-n\rangle. Write \hat{F} in a form like \hat{H} in the first equation and show that \hat{F} is both unitary and commutes with \hat{H}, thus showing that \hat{F} is also a symmetry of the Hamiltonian.

(d) Find a complete set of eigenstates of \hat{F} and their associated eigenvalues. Do all these eigenstates have to be eigenstates of \hat{H} as well? If not, do any of these eigenstates have to be eigenstates of \hat{H}? Explain your answer.

Solution

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