Quantum Qual Problems

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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.