Transformations of Operators and Symmetry

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

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 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{A}} and a transformation 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{T},} we define the transformed 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{A}'} as follows. Given the relation,

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

between two vectors 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} 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\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, they 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. Moreover, 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 ${\hat {U}}:$

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{U}\hat{H}|\psi\rangle=\hat{U}|\phi\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 \hat{U}} is a symmetry of the Hamiltonian, then it must also be true 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{H}\hat{U}|\psi\rangle=\hat{U}|\phi\rangle.}

Subtracting these two equations, we see that, because 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} is arbitrary, the Hamiltonian commutes with the transformation operator; i.e., 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{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. Many of the transformations that we will consider in quantum mechanics are "generated" from observables - for example, as we saw in the previous section, the translation 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{T}(\mathbf{l})=\exp\left (-\frac{i\hat{\mathbf{p}}\cdot\mathbf{l}}{\hbar}\right ),} is "generated" from the momentum. In fact, any unitary transformation can be written in the form, 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{U}=e^{i\hat{O}},} where 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{O}} is a (dimensionless) Hermitian operator, and thus can be said to be "generated" from an observable. If such a transformation commutes with the Hamiltonian, then it implies that the "generating" observable itself commutes with the Hamiltonian, and thus the eigenstates of the Hamiltonian are also eigenstates of the observable. This allows us to classify our eigenstates according to the value of the given observable.

Commutators & symmetry

We can define an operator called the parity 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{P}} which does the following:

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

The parity operator commutes with the 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 \hat{H}} if the potential is symmetric, 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{V}(r)=\hat{V}(-r)} . 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, the solutions can be chosen to have definite parity (even and odd functions).

Problem

Solution