Difference between revisions of "Quantum Qual Problems"

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*  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.
 
*  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.
  
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Solution: [[Qal_QuantP1S]]
  
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Solution:
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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
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<math>\begin{pmatrix}1 \\ 0\end{pmatrix}</math>
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and
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<math>\begin{pmatrix} 0 \\ 1\end{pmatrix}</math>.
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The Hamiltonian can be written as
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<math>H_0= \begin{pmatrix} w_1 & 0 \\ 0 & w_2\end{pmatrix}</math>
  
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a.) We introduce and interaction <math>H_1</math> whose matrix elements, using the above basis vectors, are
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<math>H_1= \begin{pmatrix} 0 & v \\ v & 0\end{pmatrix}</math>
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where v is real.
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Find the exact values of the energies of the new Hamitonlian, <math>H=H_0 + H_1</math>
  
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2.)
 
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  Dr. Forest h is "h bar"<br>
 
  
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b.) Assume that <math>w_1 - w_2 >> |v|</math>.  Use time independent perturbation theory to compute the first and second order corrections to the energy levels of <math>H_0</math> when <math>H_1</math> is treated as a perturbation. 
  
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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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c.) Let <math>w=w_1=w_2</math> in <math>H_0</math> above. Assuming the system is initially in the eigenstate
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<math>\begin{pmatrix}1 \\ 0\end{pmatrix}</math> of <math>H_0</math>, and the interaction <math>H_1</math> is turned on for a finite time interval <math>T</math>.  Find the probabitlity that the system will be found in the state <math>H_0= \begin{pmatrix} w_1 & 0 \\ 0 & w_2\end{pmatrix}</math> at the end of that time interval.
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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}{1}</math>
 

Latest revision as of 21:13, 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.

Solution: 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]\begin{pmatrix}1 \\ 0\end{pmatrix}[/math] and [math]\begin{pmatrix} 0 \\ 1\end{pmatrix}[/math]. The Hamiltonian can be written as [math]H_0= \begin{pmatrix} w_1 & 0 \\ 0 & w_2\end{pmatrix}[/math]

a.) We introduce and interaction [math]H_1[/math] whose matrix elements, using the above basis vectors, are [math]H_1= \begin{pmatrix} 0 & v \\ v & 0\end{pmatrix}[/math] where v is real. Find the exact values of the energies of the new Hamitonlian, [math]H=H_0 + H_1[/math]


b.) Assume that [math]w_1 - w_2 \gt \gt |v|[/math]. Use time independent perturbation theory to compute the first and second order corrections to the energy levels of [math]H_0[/math] when [math]H_1[/math] is treated as a perturbation.

c.) Let [math]w=w_1=w_2[/math] in [math]H_0[/math] above. Assuming the system is initially in the eigenstate [math]\begin{pmatrix}1 \\ 0\end{pmatrix}[/math] of [math]H_0[/math], and the interaction [math]H_1[/math] is turned on for a finite time interval [math]T[/math]. Find the probabitlity that the system will be found in the state [math]H_0= \begin{pmatrix} w_1 & 0 \\ 0 & w_2\end{pmatrix}[/math] at the end of that time interval.

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