Difference between revisions of "Quantum Qual Problems"

From New IAC Wiki
Jump to navigation Jump to search
Line 13: Line 13:
 
and  
 
and  
 
<math>\begin{pmatrix} 0 \\ 1\end{pmatrix}</math>
 
<math>\begin{pmatrix} 0 \\ 1\end{pmatrix}</math>
 
 
 
 
 
the Hamiltonian can be written as
 
the Hamiltonian can be written as
 +
<math>H_0= \begin{pmatrix} w_1 & 0 \\ 0 & w_2\end{pmatrix}</math>

Revision as of 21:07, 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]