Difference between revisions of "Quantum Qual Problems"

From New IAC Wiki
Jump to navigation Jump to search
 
(49 intermediate revisions by the same user not shown)
Line 7: Line 7:
 
*  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: [[Qal_QuantP1S]]
  
−
Solution:
+
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
 +
<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>
  
−
2.)
 
−
* <math> [- \frac{h^2}{2m}\Delta^2 + V]W(x,y,z)=E W(x,y,z) </math><br>
 
  
−
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>
+
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. 
  
−
<math>\frac{d^2 w(x)}{dx^2} - c^2 w(x) = 0</math>(1)<br>
+
c.) Let <math>w=w_1=w_2</math> in <math>H_0</math> above.  Assuming the system is initially in the eigenstate
−
The same will be for y and z.<br>
+
<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.
−
 
 
−
Solution of equation (1) is following <br>
 
−
<math>w(x) = A\sin(cx)+B\cos(cx)</math><br>
 
−
<math>w(y) = C\sin(cy)+D\cos(cy)</math><br>
 
−
<math>w(z) = E\sin(cz)+F\cos(cz)</math><br>
 
−
 
 
−
* Applying B. C. at x=y=z=0 wave function should be zero, that means B=D=F=0. We have<br>
 
−
 
 
−
<math>w(x) = A\sin(cx) </math><br>
 
−
<math>w(y) = C\sin(cy) </math><br>
 
−
<math>w(z) = E\sin(cz) </math><br>
 

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.

Retrieved from "https://wiki.iac.isu.edu/index.php?title=Quantum_Qual_Problems&oldid=22107"