Time Evolution of Stationary States

Classical Mechanics Level 2

A particle in quantum mechanics has the non-normalized wavefunction at t = 0 t=0 :

ψ ( x ) = − i ϕ 1 + ϕ 3 , \psi(x) = -i \phi_1 + \phi_3,

where the ϕ n \phi_n are the orthonormal eigenstates of some Hamiltonian with energies E n = 1 n E_n = \frac{1}{n} .

Find the normalized wavefunction ψ ( x , t ) \psi(x,t) at all times.

− i e − i t / ℏ ϕ 1 + e − i t / ( 3 ℏ ) ϕ 3 -i e^{-it/\hbar} \phi_1 + e^{-it/(3\hbar)} \phi_3 − i 2 ϕ 1 + 1 2 ϕ 3 -\frac{i}{\sqrt{2}} \phi_1 + \frac{1}{\sqrt{2}} \phi_3 − i 2 e − i t / ℏ ϕ 1 + 1 2 e − 3 i t / ℏ ϕ 3 -\frac{i}{\sqrt{2}} e^{-it/\hbar} \phi_1 + \frac{1}{\sqrt{2}} e^{-3it/\hbar} \phi_3 − i 2 e − i t / ℏ ϕ 1 + 1 2 e − i t / ( 3 ℏ ) ϕ 3 -\frac{i}{\sqrt{2}} e^{-it/\hbar} \phi_1 + \frac{1}{\sqrt{2}} e^{-it/(3\hbar)} \phi_3

Matt DeCross by Matt DeCross , 27, USA

This section requires Javascript.
You are seeing this because something didn't load right. We suggest you, (a) try refreshing the page, (b) enabling javascript if it is disabled on your browser and, finally, (c) loading the non-javascript version of this page . We're sorry about the hassle.

1 solution

Matt DeCross
May 10, 2016

Each energy eigenstate evolves according to its eigenstate by the factor e − i E n t / ℏ = e − i t / ( ℏ n ) e^{-iE_n t/\hbar} = e^{-i t/(\hbar n)} .

First, normalize the state by applying a factor of 1 2 \frac{1}{\sqrt{2}} to each term so that the total norm is one. Then, evolving ϕ 1 \phi_1 by e − i t / ℏ e^{-it / \hbar} and ϕ 3 \phi_3 by e − i t / ( 3 ℏ ) e^{-it / (3\hbar)} gives the solution.

1 Helpful 0 Interesting 0 Brilliant 1 Confused

0 pending reports

×

Problem Loading...

Note Loading...

Set Loading...