IHateMayonnaise
Oct1-10, 11:38 AM
1. The problem statement, all variables and given/known data
This is Exercise 9 from chapter 15 of merzbacer. It asks to find \lvert\psi(x,t)\rvert^2 given:
\psi(x,t)=[2\pi(\Delta x)_0^2]^{-1/4}\left[1+\frac{i\hbar t}{2m(\Delta x)_0^2} \right]^{-1/2} \exp\left[\frac{-\frac{x^2}{4(\Delta x)_0^2}+ik_0x-ik_0^2\frac{\hbar t}{2m}}{1+\frac{i\hbar t}{2m(\Delta x)_0^2}}\right]
2. Relevant equations
\lvert\psi(x,t)\rvert^2=\psi^*(x,t)\psi(x,t)
(\Delta x)^2=\langle x \rangle^2 - \langle x^2\rangle
i\hbar\frac{d}{dt}\langle A \rangle = \langle[A,H]\rangle + \left\langle\frac{\partial A}{\partial t} \right\rangle
3. The attempt at a solution
My question is quick and qualitative: is there..an easier way of doing this than the brute force way? I mean, am I not seeing something? Or is this problem as useless as it seems?
I am not allowed to use a computer in any way.
This is Exercise 9 from chapter 15 of merzbacer. It asks to find \lvert\psi(x,t)\rvert^2 given:
\psi(x,t)=[2\pi(\Delta x)_0^2]^{-1/4}\left[1+\frac{i\hbar t}{2m(\Delta x)_0^2} \right]^{-1/2} \exp\left[\frac{-\frac{x^2}{4(\Delta x)_0^2}+ik_0x-ik_0^2\frac{\hbar t}{2m}}{1+\frac{i\hbar t}{2m(\Delta x)_0^2}}\right]
2. Relevant equations
\lvert\psi(x,t)\rvert^2=\psi^*(x,t)\psi(x,t)
(\Delta x)^2=\langle x \rangle^2 - \langle x^2\rangle
i\hbar\frac{d}{dt}\langle A \rangle = \langle[A,H]\rangle + \left\langle\frac{\partial A}{\partial t} \right\rangle
3. The attempt at a solution
My question is quick and qualitative: is there..an easier way of doing this than the brute force way? I mean, am I not seeing something? Or is this problem as useless as it seems?
I am not allowed to use a computer in any way.