10 - Collapse of Wave Function

Collapse of Wave Function

Let $A$ be an observable

Upon measuring observable $A$ and getting non-degenerate eigenvalue ${\alpha }_n$ then

Immediately after measuring quantum state of system is eigenstate ${\psi }_n\in \mathcal{H}$ with

$$A{\psi }_n={\alpha }_n{\psi }_n$$

Collapse of Wave Function - Degenerate Case

Let $A$ be an observable

If spectrum of observable $A$ has degenerate eigenvalues then
Quantum State $\psi$ as expansion from degeneracy of an eigenvalue
Immediately after measuring eigenvalue ${\alpha }_n$ then quantum state of system is

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

Deterministic and Probabilistic Process of Time Evolution

(Deterministic)
Normalised quantum state of system $\psi (t)$ evolves in time according to Schrödinger Equation
Given any observable $A$ then

$$\psi (t)=\sum _n{c}_n(t){\psi }_n$$

where $\{{\psi }_n\}$ are the complete orthonormal eigenstates of $A$
Typically chosen that $A=H$ so ${\psi }_n$ are stationary states in $E_n$ and

$$c_n(t)=c_n{e}^{-i{E}_{n}t/\hslash }\text{ where }{c}_n\in \mathbb{C}\text{ is constant in time}$$

(Probabilistic)
Suppose at time $t=t_0$ then observable $A$ is measured then

$${\mathbb{P}}_{\psi }(\text{obtaining eigenvalue }{\alpha }_n\text{ of }A)=|⟨{\psi }_n\mid \psi (t_0)⟩{|}^2$$

where ${\psi }_n$ is the eigenstate of $A$ with eigenvalue ${\alpha }_n$

By Collapse of Wave Function then immediately after obtaining value ${\alpha }_n$ then
Wave function collapses to $\psi (t_0)\to {\psi }_n$
Now start time evolution again using Schrödinger Equation with initial condition

$$\psi (t_0)={\psi }_n$$