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]

a.) We introduce and interaction [math]\displaystyle{ H_1 }[/math] whose matrix elements, using the above basis vectors, are [math]\displaystyle{ 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]\displaystyle{ H=H_0 + H_1 }[/math]

b.) Assume that [math]\displaystyle{ 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]\displaystyle{ H_0 }[/math] when [math]\displaystyle{ H_1 }[/math] is treated as a perturbation.

c.) Let [math]\displaystyle{ w=w_1=w_2 }[/math] in [math]\displaystyle{ H_0 }[/math] above. Assuming the system is initially in the eigenstate [math]\displaystyle{ \begin{pmatrix}1 \\ 0\end{pmatrix} }[/math] of [math]\displaystyle{ H_0 }[/math], and the interaction [math]\displaystyle{ H_1 }[/math] is turned on for a finite time interval [math]\displaystyle{ T }[/math]. Find the probabitlity that the system will be found in the state [math]\displaystyle{ H_0= \begin{pmatrix} w_1 & 0 \\ 0 & w_2\end{pmatrix} }[/math] at the end of that time interval.