- #1
TFM
- 1,026
- 0
Homework Statement
Consider a particle that is confined in a one-dimensional box, ie in a potential
[tex] V(x) = 0 for 0 \leq x \leq L, or \infty when x < 0, x > L[/tex]
(i) Determine the solutions Φn(x) of the stationary Schrödinger equation for this problem. Make sure that you have normalized them correctly.
(ii) Calculate the energy eigenvalue En corresponding to Φn(x).
(iii) Use the results of (i) and (ii) and write down the complete time-dependent wave function Ψn(x,t) for the nth stationary state in this potential.
( iv) For the nth stationary state calculate .
(v) Use the results of (iv) to check whether the Heisenberg Uncertainty relation is satisfied for the nth stationary state? Which state comes closest to the minimum uncertainty?
Homework Equations
The Attempt at a Solution
Okay, I have done (i), and have got:
[tex] k = \frac{n\pi}{L} [/tex]
and
[tex] \phi(x) = Asin(\frac{n\pi}{L}x) [/tex]
Where n is plus/minus 1,2,3...
Okay, but I am not sure on the second part.
I have a book, and it has:
[tex] E_n = \frac{\hbar^2k_n^2}{2m} [/tex]
And I think this is the irght formula. The trouble is that IO believe that this equation isbasically the answer, just sub in the result I got above, but I don't know where the equation has come from. Could anyone help me out here?
Thanks in Advance,
TFM