Difference between revisions of "Quantum Qual Problems"

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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 x < a, 0< y< b </math> and <math>0<z<c</math>.
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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 <x < a, 0< y< b </math> and <math>0<z<c</math>.
  
 
* Use the time-independent Schrodinger equation for this problem to obtain the general form for the eigenfunctions of the particle
 
* Use the time-independent Schrodinger equation for this problem to obtain the general form for the eigenfunctions of the particle
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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.
  
Solution:
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Solution: [[Qal_QuantP1S]]
  
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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>
  
2.)
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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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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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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.

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.