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.
  
[[Qal_QuantP1S]]
+
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 > 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  
  
<math>\[ \left (  
+
<math>\[ \left ( \begine{array} {c} 1 \\ 0 \end{array} \right ) </math>
\begine{array} {c}
 
1 \\
 
0
 
\end{array} \right)
 
 
 
</math>
 
 
the Hamiltonian can be written as
 
the Hamiltonian can be written as

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.

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