Quantum Qual Problems

1.) Given a quantum mechanical particle of mass [math]\displaystyle{ M }[/math] confined inside a box of sides [math]\displaystyle{ a,b,c }[/math]. The particle is allowed to move freely between [math]\displaystyle{ 0 <x < a, 0< y< b }[/math] and [math]\displaystyle{ 0<z<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]\displaystyle{ a=b=c }[/math] and find the first 6 eigenenergies of the problem in terms of the box side length ([math]\displaystyle{ a }[/math]), the particle mass ([math]\displaystyle{ 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]\displaystyle{ w_1 }[/math] and [math]\displaystyle{ w_2 }[/math]. Assume that [math]\displaystyle{ w_1 > w_2 }[/math]. Representing the enegy eigenstate by [math]\displaystyle{ \begin{pmatrix}1 \\ 0\end{pmatrix} }[/math] and [math]\displaystyle{ \begin{pmatrix} 0 \\ 1\end{pmatrix} }[/math] the Hamiltonian can be written as [math]\displaystyle{ H_0= \begin{pmatrix} w_1 & 0 \\ 0 & w_2\end{pmatrix} }[/math]