(adsbygoogle = window.adsbygoogle || []).push({}); [SOLVED] Calculate time evolution of Schrodinger wave equation

1. The problem statement, all variables and given/known data

At time t=0 particle is in state:

[tex]\psi\left(x\right)=\sqrt{2}A\phi_{1}(x)+\frac{A}{\sqrt{2}}\phi_{2}(x)+A\phi_{3}(x)[/tex]

where [tex]\phi_{n}(x)[/tex] are eigenfunctions of 1-D infinite potential well.

a) Normalize the state

b) calculate [tex]\psi(x,t)[/tex]

d) Calculate <E>

2. Relevant equations

[tex]\psi\left(x,t\right)=\sum_{n}b_{n}*\phi_{n}(x)*e^{\frac{-iE_{n}t}{h/2Pi}}[/tex]

[tex]b_{n}=<\phi_{n}(x)|\psi(x,0)>[/tex] [tex]\Rightarrow[/tex] [tex]b_{n}=\int^{L}_{0}\sqrt{\frac{2}{L}}*Sin(\frac{n*Pi*x}{L})*\psi(x,0)*dx[/tex]

3. The attempt at a solution

From normalization I've got A = (2/7)^1/2.

To calculate [tex]\psi(x,t)[/tex] I have used

[tex]\psi\left(x,t\right)=\sum_{n}b_{n}*\phi_{n}(x)*e^{\frac{-iE_{n}t}{h/2Pi}}[/tex]

To get to bn coefficients I need to calculate the integral:

[tex]b_{n}=\int^{L}_{0}\sqrt{\frac{2}{L}}*Sin(\frac{n*Pi*x}{L})*\psi(x,0)*dx[/tex]

This integral gives Sin[n*Pi] multiplied by some expression,

but this means thath all bn coefficients are zero (since Sin[n*Pi]=0 for every integer n).

So my [tex]\psi(x,t)[/tex] is always zero except for t=0, is this possible ?

This also means that probability of finding particle later inside the well is zero,

but particle should be located shomewere inside well, potential walls are infinite so particle

can't escape !?

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# Homework Help: Calculate time evolution of Schrodinger wave equation

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