- #1
frerk
- 19
- 1
hey,
this thread might be a bit longer, but I have already calculated everything and I am quiet sure that it is right :)
Just need someone who confirms me or not :)
1. Homework Statement
Time evolution of probabilities. An electron inside a quantum well of length L is at time t = 0 in the state
[tex] \psi(x,0) =\frac{1}{ \sqrt{L} } ( \sin( \frac{\pi x}{ L} ) + \sin( \frac{2 \pi x}{L} )) [/tex]
Outside the well [tex] \psi(x,0) = 0. [/tex] The state [tex] \psi(x,0) [/tex] is thus a superposition of two stationary states[tex] \psi_{1}(x) \ and \ \psi_{2}(x) [/tex] with the energies [tex] E_{1} = \frac { \pi^2\hbar^2 }{2mL^2} \ and \ E_{2} = 4 E_{1} [/tex]
Calculate the porbability that the electron is in the interval (0, L/2) at the time [tex] t = \pi \hbar / E_{1} [/tex]
So, that is the homework, now I show you how i get a result:
[tex] \psi(x,t) = \psi(x,0) e^{-iHt/\hbar} [ \psi_1(x,0) + \psi_2(x,0) ] e^{-iHt/\hbar} =\psi_1(x,0)e^{-iE_1 t/\hbar } + \psi_2(x,0)e^{-iE_2 t/\hbar } [/tex]
[tex] \Rightarrow | \psi(x,t)|^2 = |\psi_1(x,0)e^{-iE_1 t/\hbar } + \psi_2(x,0)e^{-iE_2 t/\hbar }|^2 [/tex]
[tex] = | \psi_1(x,0)|^2 + |\psi_2(x,0)|^2 + \psi_1(x,0)^* \psi_2(x,0) e^{i(E_1 - E_2) t / \hbar } + \psi_2(x,0)^* \psi_1(x,0) e^{i ( E_2 -E_1) t / \hbar} [/tex]
Now we put this into the integral:
[tex]
\int_0^b \! | \psi(x,t)|^2 \, d x [/tex] where b = L/2 and we geht the result
[tex] 0,5 + \frac{2}{3\pi} e^{i(E_1 E_2)t/\hbar} + \frac{2}{3\pi} e^{-i(E_1 E_2)t/\hbar} [/tex]
Now I insert: [tex] t = \pi\hbar/E_1 = \pi \hbar 2mL^2 / \pi^2 \hbar^2 = 2mL^2/(\pi\hbar) [/tex]
[tex] E_1 = \pi^2\hbar^2 / (2mL^2) [/tex]
[tex] E_2 = 4E_1 = 2\pi^2 \hbar^2 / (mL^2) [/tex]
And after doing that, we find out that the probability that the electron in in the interval (0,L/2) for the specific time is 0,076.
Thank you :)
this thread might be a bit longer, but I have already calculated everything and I am quiet sure that it is right :)
Just need someone who confirms me or not :)
1. Homework Statement
Time evolution of probabilities. An electron inside a quantum well of length L is at time t = 0 in the state
[tex] \psi(x,0) =\frac{1}{ \sqrt{L} } ( \sin( \frac{\pi x}{ L} ) + \sin( \frac{2 \pi x}{L} )) [/tex]
Outside the well [tex] \psi(x,0) = 0. [/tex] The state [tex] \psi(x,0) [/tex] is thus a superposition of two stationary states[tex] \psi_{1}(x) \ and \ \psi_{2}(x) [/tex] with the energies [tex] E_{1} = \frac { \pi^2\hbar^2 }{2mL^2} \ and \ E_{2} = 4 E_{1} [/tex]
Calculate the porbability that the electron is in the interval (0, L/2) at the time [tex] t = \pi \hbar / E_{1} [/tex]
The Attempt at a Solution
So, that is the homework, now I show you how i get a result:
[tex] \psi(x,t) = \psi(x,0) e^{-iHt/\hbar} [ \psi_1(x,0) + \psi_2(x,0) ] e^{-iHt/\hbar} =\psi_1(x,0)e^{-iE_1 t/\hbar } + \psi_2(x,0)e^{-iE_2 t/\hbar } [/tex]
[tex] \Rightarrow | \psi(x,t)|^2 = |\psi_1(x,0)e^{-iE_1 t/\hbar } + \psi_2(x,0)e^{-iE_2 t/\hbar }|^2 [/tex]
[tex] = | \psi_1(x,0)|^2 + |\psi_2(x,0)|^2 + \psi_1(x,0)^* \psi_2(x,0) e^{i(E_1 - E_2) t / \hbar } + \psi_2(x,0)^* \psi_1(x,0) e^{i ( E_2 -E_1) t / \hbar} [/tex]
Now we put this into the integral:
[tex]
\int_0^b \! | \psi(x,t)|^2 \, d x [/tex] where b = L/2 and we geht the result
[tex] 0,5 + \frac{2}{3\pi} e^{i(E_1 E_2)t/\hbar} + \frac{2}{3\pi} e^{-i(E_1 E_2)t/\hbar} [/tex]
Now I insert: [tex] t = \pi\hbar/E_1 = \pi \hbar 2mL^2 / \pi^2 \hbar^2 = 2mL^2/(\pi\hbar) [/tex]
[tex] E_1 = \pi^2\hbar^2 / (2mL^2) [/tex]
[tex] E_2 = 4E_1 = 2\pi^2 \hbar^2 / (mL^2) [/tex]
And after doing that, we find out that the probability that the electron in in the interval (0,L/2) for the specific time is 0,076.
Thank you :)
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