Transformations of Operators and Symmetry

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.

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

Symmetry of any quantum mechanical state is determined by how the state transforms under certain mathematical transformations, examples being translation and rotation. A symmetry transformation is a transformation that keeps the physical characteristics of the system unchanged (for example, a rotation of a spherical object). Of special importance are problems for which the Hamiltonian is left invariant under a symmetry transformation.

In addition, in both classical and quantum mechanics, symmetry transformations become important due to their relation to conserved quantities. Moreover in quantum mechanics the importance of symmetries is further enhanced by the fact that observation of conserved quantities can be exactly predictable in spite of the probabilistic nature of quantum predictions.

Let us consider an arbitrary transformation of an arbitrary state | n > to be given by the operator U such that the transformation gives | n > $\rightarrow$ U | n >

If U produce a symmetry transformation, the following theorems hold.

If the operator U produces a symmetry transformation on all ket vectors, then it must commute with the hamiltonian.

Proof: By definition of a symmetry transformation, the operator U could transform an energy eigenstate either to itself or another eigenstate degenerate to it. Hence, if | E_i > is an eigenstate of H with eigenvalue Ei then

$HU|E_{i}>=HU|E_{i}^{'}>=E_{i}|E_{i}^{'}>=E_{i}U|E_{i}>=UE_{i}|E_{i}>=UH|E_{i}>$

$|E_{i}>$ and $|E_{i}^{'}>$

Therefore we can write,

$\left[H,U\right]|E_{i}>=0$

This is valid for all energy eigenstates $|E_{i}>.$

Now, from the completeness theorem any arbitrary state $|n>$

can be written as a linear combination of the eigenstates $|E_{i}>$ .

Hence, we can write,

$\left[H,U\right]|n>=0$

Since $|n>$

is an arbitrary ket vector, we can conclude that

$\left[H,U\right]=0$

Problem on symmetry

Commutators & symmetry

We can define an operator called the parity operator, ${\hat {P}}$ which does the following:

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

The parity operator commutes with the Hamiltonian ${\hat {H}}$ if the potential is symmetric, ${\hat {V}}(r)={\hat {V}}(-r)$. 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).