Difference between revisions of "Quantum Qual Problems"
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<math>\frac{1}{A^2}=\int\sin^2 (\pi nx/a) dx = \frac{a}{2} </math>, limits are from 0 to a. <br> | <math>\frac{1}{A^2}=\int\sin^2 (\pi nx/a) dx = \frac{a}{2} </math>, limits are from 0 to a. <br> | ||
− | The | + | The eigenfunction for each component will be<br> |
<math>w(x) = \sqrt{\frac{2}{a}} \sin(\pi n_x x/a)</math><br> | <math>w(x) = \sqrt{\frac{2}{a}} \sin(\pi n_x x/a)</math><br> | ||
<math>w(x) = \sqrt{\frac{2}{b}} \sin(\pi n_y y/b)</math><br> | <math>w(x) = \sqrt{\frac{2}{b}} \sin(\pi n_y y/b)</math><br> | ||
<math>w(x) = \sqrt{\frac{2}{c}} \sin(\pi n_z z/c)</math><br> | <math>w(x) = \sqrt{\frac{2}{c}} \sin(\pi n_z z/c)</math><br> | ||
+ | |||
+ | The eigenenergies <br> | ||
+ | |||
+ | <math>\E_n_x</math> |
Revision as of 03:35, 16 August 2007
1.) Given a quantum mechanical particle of mass
confined inside a box of sides . The particle is allowed to move freely between and .- Use the time-independent Schrodinger equation for this problem to obtain the general form for the eigenfunctions of the particle
- Now apply boundary conditions to obtain the specific eigenfunctions and eigenenergies for this specific problem.
- Assume and find the first 6 eigenenergies of the problem in terms of the box side length ( ), the particle mass ( ) and standard constants. What are their quantum number? Make a sketch of the eigenvalue spectrum, a table listing these eigenenergies and the quantum numbers of all the states that correspond to them.
Solution:
2.)
In our case, using separation of variables, we will get 3 differential equations for x, y and z. W(x,y,z)=w(x)w(y)w(z)
The same will be for y and z.
Solution of equation (1) is following
- Applying B. C. at x=y=z=0 wave function should be zero, that means B=D=F=0. We have
Also, w(a)=0 which gives
A, C and E are normalization constants
The eigenfunction for each component will be
The eigenenergies