09 - Eigenfunctions and Eigenstates

Eigenstate

Let $A$ be an observable (self-adjoint linear operator on $\mathcal{H}$) then

Let state $\psi \in \mathcal{H}$ satisfy

$$A\psi =\alpha \psi$$

where $a\in \mathbb{C}$ is a constant

Then

$$\psi \text{ is an eigenstate of }A\text{ with eigenvalue }\alpha$$

Spectrum

Spectrum is the set of all eigenvalues of $A$

Realness of Eigenvalues of Observables

Let $\alpha$ be an eigenvalue of observable $A$ then

$$\alpha \text{ is real }$$

Proof

Consider $\alpha ⟨\psi \mid \psi ⟩$ then

$$\alpha ⟨\psi \mid \psi ⟩=⟨\psi \mid \alpha \psi ⟩=⟨\psi \mid A\psi ⟩=⟨A\psi \mid \psi ⟩=⟨\alpha \psi \mid \psi ⟩=\overline{\alpha }⟨\psi \mid \psi ⟩$$

As $⟨\psi \mid \psi ⟩\ne 0$ then $\alpha =\overline{\alpha }$ so $\alpha$ is real

Orthogonality of Eigenstates of Distinct Eigenvalues

Suppose ${\psi }_1,{\psi }_2$ are eigenstates with distinct eigenvalues ${\alpha }_1\ne {\alpha }_2$

Then

$$⟨{\psi }_2\mid {\psi }_1⟩=0\text{ if }{\alpha }_1\ne {\alpha }_2$$

So ${\psi }_1$ and ${\psi }_2$ are orthogonal

Proof

As $A{\psi }_1={\alpha }_1{\psi }_1$ and $A{\psi }_2={\alpha }_2{\psi }_2$ then

Consider $⟨{\psi }_2\mid {\psi }_1⟩$

$$\begin{array}{rl}{\alpha }_1⟨{\psi }_2\mid {\psi }_1⟩& =⟨{\psi }_2\mid {\alpha }_1{\psi }_1⟩\\ & =⟨{\psi }_2\mid A{\psi }_1⟩\\ & =⟨A{\psi }_2\mid {\psi }_1⟩\\ & =⟨{\alpha }_2{\psi }_2\mid {\psi }_1⟩\\ & =\overline{{\alpha }_2}⟨{\psi }_2\mid {\psi }_1⟩\\ & ={\alpha }_2⟨{\psi }_2\mid {\psi }_1⟩\end{array}$$

As ${\alpha }_1\ne {\alpha }_2$ then $⟨{\psi }_2\mid {\psi }_2⟩=0$

Complete Set of Eigenstates for Observables axiom

Observable $A$ is required to have a complete set of eigenstates so
There exists a set of orthonormal eigenstates $\{{\psi }_n\}$ of $A$ where

$$A{\psi }_n={\alpha }_n{\psi }_n\text{ with }{a}_n\in \mathbb{R}\text{ and }⟨{\psi }_m\mid {\psi }_n⟩={\delta }_{mn}$$

Hence for any $\psi \in \mathcal{H}$ then it can be written as

$$\psi =\sum _n{c}_n{\psi }_n$$

where $c_n\in \mathbb{C}$ and $c_n=⟨{\psi }_n\mid \psi ⟩$

Note that if $\psi =\psi (t)$ then $c_n=c_n(t)$ (dependence on time)

Degenerate Form $\{a_n\}$ be the distinct eigenvalues where $\{{\psi }_{n,i}\}$ form an orthonormal basis for eigenspace with eigenvalue ${\alpha }_n$

Let

Degeneracy of an Eigenvalue

Consider Eigenspace of a Eigenvalue then

Dimension $d$ of Eigenspace is the Degeneracy of Eigenvalue

Note

$$\psi =\sum _n\sum _{i=1}^{d_n}{c}_{n,i}{\psi }_{n,1}$$

where $d_n$ is degeneracy of eigenvalue ${\alpha }_n$ and $c_{n,i}\in \mathbb{C}$ and
${\psi }_{n,i}$ form a orthonormal basis for eigenspace with eigenvalue ${\alpha }_n$

Non-Degenerate Eigenvalue

Eigenvalue $\alpha$ is non-degenerate if degeneracy $d=1$

Quantum Measurement Postulate

Possible Measurement Outcomes of an observable $A$ are the eigenvalues $\{{\alpha }_n\}$ of $A$

Suppose system is in a normalised quantum state $\psi$ in form

If eigenvalue ${\alpha }_n$ is non-degenerate then probability of obtaining ${\alpha }_n$ in a measurement is

$${\mathbb{P}}_{\psi }(\text{obtaining }{\alpha }_n)=|c_n{|}^2=|⟨{\psi }_n\mid \psi ⟩{|}^2$$

If eigenvalue ${\alpha }_n$ is degenerate $d_n>1$ then

$${\mathbb{P}}_{\psi }(\text{obtaining }{\alpha }_n)=\sum _{i=1}^{d_n}|c_{n,i}{|}^2=\sum _{i=1}^{d_n}|⟨{\psi }_{n,i}\mid \psi ⟩{|}^2$$

Orthogonality of the Hermite Polynomials

$${\int }_{-\infty }^{\infty }{H}_n(\xi )H_m(\xi )e^{-{\xi }^2}d\xi =0\text{ for }m\ne n$$

Any normalisable function $\psi$ on $\mathbb{R}$ can be written in an expansion in eigenstates of $H$

$$\psi (\xi )=\sum _{n=0}^{\infty }{c}_n{H}_n(\xi )e^{-{\xi }^2/2}$$