Physics 312 Problem Set 2
1. The simple harmonic oscillator and the general variational method.
(a) In class we proposed finding the ground state solution of the quantum mechanical SHO by using the
variational method and choosing the trial wavefunction φ = α2 − x2 for |x| < α; φ = 0 for |x| > α. Find
the best ground state energy and associated wavefunction this choice provides.
(b) Choose for the ground state energy the trial wavefunction
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φ = Ce−γx ,
where γ is the parameter to be found and C is the normalization constant. Find the ground state eigenvalue
and eigenfunction.
(c) Choose for the first excited state the trial wavefunction
φ = C1 xe−γ1 x
2
and find the first excited state eigenvalue and eigenfunction.
2. Anharmonic (x4 ) oscillator and the linear variational method.
A one dimensional SHO is subjected to the modification H (1) = βx4 . Choosing the space of basis functions to
be the unperturbed SHO ground states |0i and |2i, find
(a) The best estimate for the actual energy eigenvalues under the modification, and
(b) Their associated eigenvectors.
3. C-TDL (Cohen-Tannoudji) Chapter 11, Complement HXI Problem 3.
4. C-TDL HXI Problem 8.
5. Working out the spectrum from commutation relations. The generators of SU(2) can be chosen to fulfill the
following commutations relations:
[Ji , Jj ] = iijk Jk , i, j, k = 1, 2, 3 .
We can also, without loss of generality, choose to express the eigenvalues and eigenvectors of J~2 and Jz as
follows:
J~2 |jmi = h̄2 j(j + 1)|jmi
Jz |jmi = h̄m|jmi
All that remains is to find the possible values of j and m. You will also need that J± = Jx ± iJy .
(a) Use the value of hψ|J~2 |ψi to argue that you may choose j ≥ 0.
(b) Use the value of |J+ |jmi|2 to argue that either j ≥ m ≥ −j − 1 or j ≤ m ≤ −j − 1
(c) Use the value of |J− |jmi|2 to argue that either j + 1 ≥ m ≥ −j or j + 1 ≤ m ≤ −j
(d) Find the relationship between j and m that satisfies all of the above constraints.
(e) Solve for the values of m such that |J− |jmi|2 = 0 and the constraint from (d) is satisfied.
(f) Use the commutators [J~2 , J± ] and [Jz , J± ] and normalization of the states to find the eigenvalues and
normalized eigenvectors of the states J± |jmi >.
(g) Use the result of part (f) to show that 2j = n, where n is an integer.
6. C-TDL Ch. IX problem 1.
7. C-TDL Ch. IX problem 2.
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