Difference between revisions of "Quantum Qual Problems"

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[[Qal_QuantP1S]]
 
[[Qal_QuantP1S]]
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Solution:
 
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* <math> [- \frac{\hbar^2}{2M}\Delta^2 + V]W(x,y,z)=E W(x,y,z) </math><br>
 
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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)<br>
 
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<math>\frac{d^2 w(x)}{dx^2} - m^2 w(x) = 0</math>(1)<br>
 
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The same will be for y and z.<br>
 
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Solution of equation (1) is following <br>
 
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<math>w(x) = A\sin(mx)+B\cos(mx)</math><br>
 
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<math>w(y) = C\sin(ky)+D\cos(ky)</math><br>
 
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<math>w(z) = E\sin(qz)+F\cos(qz)</math><br>
 
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* Applying B. C. at x=y=z=0 wave function should be zero, that means B=D=F=0. We have<br>
 
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<math>w(x) = A\sin(mx) </math><br>
 
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<math>w(y) = C\sin(ky) </math><br>
 
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<math>w(z) = E\sin(qz) </math><br>
 
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Also, w(a)=0 which gives <math>A\sin(ma)=0, m=\frac{\pi n_x}{a}</math>. For y component  <math>C\sin(kb)=0, k=\frac{\pi n_y}{b}</math> and for z  <math>E\sin(qc)=0, q=\frac{\pi n_z}{c}</math><br>
 
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A, C and E are normalization constants<br>
 
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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>
 
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The eigenfunction for each component will be<br>
 
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<math>w(x) = \sqrt{\frac{2}{a}} \sin(\pi n_x x/a)</math><br>
 
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<math>w(y) = \sqrt{\frac{2}{b}} \sin(\pi n_y y/b)</math><br>
 
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<math>w(z) = \sqrt{\frac{2}{c}} \sin(\pi n_z z/c)</math><br>
 
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The eigenenergies <br>
 
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<math>E_{n_x} = \frac{\pi^2 \hbar^2 {n_x}^2}{2Ma^2}</math>, <math>E_{n_y} = \frac{\pi^2 \hbar^2 {n_y}^2}{2Mb^2}</math>, <math>E_{n_z} = \frac{\pi^2 \hbar^2 {n_z}^2}{2Mc^2}</math> <br>
 
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Total energy is sum of these energies.<br>
 
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*  <math>E = \frac{\pi^2 \hbar^2 n^2}{2M^2 a^2}</math>, where <math>n^2=n_x ^2 + n_y ^2 + n_z ^2</math>,  n=1,2,3...
 
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2.) A system has two energy eigenstate with eigenvalues <math>w_1</math> and <math>w_2</math>.  Assume that <math>w_1 > w_2</math>.  Representing the enegy eigenstate by  
 
2.) A system has two energy eigenstate with eigenvalues <math>w_1</math> and <math>w_2</math>.  Assume that <math>w_1 > w_2</math>.  Representing the enegy eigenstate by  
  

Revision as of 21:01, 17 August 2007

1.) Given a quantum mechanical particle of mass [math]M[/math] confined inside a box of sides [math]a,b,c[/math]. The particle is allowed to move freely between [math]0 \lt x \lt a, 0\lt y\lt b [/math] and [math]0\lt z\lt c[/math].

  • 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 [math]a=b=c[/math] and find the first 6 eigenenergies of the problem in terms of the box side length ([math]a[/math]), the particle mass ([math]M[/math]) 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.

Qal_QuantP1S 2.) A system has two energy eigenstate with eigenvalues [math]w_1[/math] and [math]w_2[/math]. Assume that [math]w_1 \gt w_2[/math]. Representing the enegy eigenstate by

[math]\[ \left ( \begine{array} {c} 1 \\ 0 \end{array} \right) [/math] the Hamiltonian can be written as

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