- #1

Haorong Wu

- 413

- 89

- Homework Statement
- Suppose ##\psi _p \left ( x \right ) =\frac 1 {\sqrt {2 \pi \hbar}} e^{ip_0 x/ \hbar}##, and ##H=\frac {p^2} {2m}##. The energy of the state is ##\frac {p_0^2} {2m}##. if ##\psi \left ( x,0 \right ) = \psi _p \left ( x \right )##, then calculate ##\psi \left ( x , t \right)##.

- Relevant Equations
- ##\psi \left ( x, t \right ) = e^{-iEt/\hbar} \psi \left ( x,0 \right )##

I use the equation

##\psi \left ( x, t \right ) = e^{-iEt/\hbar} \psi \left ( x,0 \right )## to calculate ##\psi \left ( x , t \right)##, and the result is ##\psi \left ( x , t \right) = \frac 1 {\sqrt {2 \pi \hbar}} exp \left [ \frac {ip_0 x} {\hbar} - \frac {i p^2 t} {2m \hbar} \right ]##.

However, it is wrong, and in the solution, first, the wave function is converted to ##\phi \left ( p \right )=\frac 1 {\sqrt {2 \pi \hbar}} \int e^{ip_0 x /\hbar} e^{-ipx/\hbar} dx = \sqrt {2 \pi \hbar} \delta \left ( p-p_0 \right ) ##.

Then, ##\psi \left ( x , t \right) = \frac 1 {\sqrt {2 \pi \hbar}} \int \sqrt {2 \pi \hbar} e^{ipx/\hbar} \delta \left ( p-p_0 \right ) dp \cdot e^{-i E t /\hbar} = e^{i p_0 x /\hbar - iEt /\hbar}##.

There is a difference of a factor of ##\frac 1 {\sqrt {2 \pi \hbar}}##, and I am confusing why it has to be converted to ##\phi \left ( p \right ) ## first?

Thanks!

##\psi \left ( x, t \right ) = e^{-iEt/\hbar} \psi \left ( x,0 \right )## to calculate ##\psi \left ( x , t \right)##, and the result is ##\psi \left ( x , t \right) = \frac 1 {\sqrt {2 \pi \hbar}} exp \left [ \frac {ip_0 x} {\hbar} - \frac {i p^2 t} {2m \hbar} \right ]##.

However, it is wrong, and in the solution, first, the wave function is converted to ##\phi \left ( p \right )=\frac 1 {\sqrt {2 \pi \hbar}} \int e^{ip_0 x /\hbar} e^{-ipx/\hbar} dx = \sqrt {2 \pi \hbar} \delta \left ( p-p_0 \right ) ##.

Then, ##\psi \left ( x , t \right) = \frac 1 {\sqrt {2 \pi \hbar}} \int \sqrt {2 \pi \hbar} e^{ipx/\hbar} \delta \left ( p-p_0 \right ) dp \cdot e^{-i E t /\hbar} = e^{i p_0 x /\hbar - iEt /\hbar}##.

There is a difference of a factor of ##\frac 1 {\sqrt {2 \pi \hbar}}##, and I am confusing why it has to be converted to ##\phi \left ( p \right ) ## first?

Thanks!