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Chapter III notes and problems for .
Some puzzling stuff in the interaction section and superposition of time-dependent states sections. Work through those here.
Problem 1. Virial Theorem.
With the assumption that is independent of time, and
I floundered with this a bit, but found the required hint in physicsforums. We can start with the Hamiltonian time derivative relation
So, with the assumption that is independent of time, and the use of a stationary state for the expectation calculation we have
The exercise now becomes one of evaluating the remaining commutators. For the Laplacian commutator we have
For the potential commutator we have
Putting all the factors back in, we get
which is the desired result.
Followup: why assume is independent of time?
Problem 2. Application of virial theorem.
Calculate with .
Problem 3. Heisenberg Position operator representation.
Express as an operator for .
We want to expand
We to evaluate to proceed. Using , we have
This gives us
Express as an operator for with .
In retrospect, for the first part of this problem, it would have been better to use the series expansion for this exponential sandwich
Or, in explicit form
Doing so, we’d find for the first commutator
so that the series has only the first two terms, and we’d obtain the same result. That seems like a logical approach to try here too. For the first commutator, we get the same result since .
The triple commutator gets no prettier, and I get
Putting all the pieces together this gives
If there is a closed form for this it isn’t obvious to me. Would a fixed lower degree potential function shed any more light on this. How about the Harmonic oscillator Hamiltonian
… this one works out nicely since there’s an even-odd alternation.
I’d not expect such a tidy result for an arbitrary potential.
Problem 4. Feynman-Hellman relation.
For continuously parametrized eigenstate, eigenvalue and Hamiltonian , and respectively, we can relate the derivatives
Left multiplication by gives
which provides the desired identity
With eigenstates and , of with eigenvalues and , respectively, and
and , determine in terms of and .
Consider a Coulomb like potential with angular momentum . If the eigenfunction is
determine , , and the energy eigenvalue in terms of , and .
We can start with the normalization constant by integrating
To go further, we need the Hamiltonian. Note that we can write the Laplacian with the angular momentum operator factored out using
With zero for the angular momentum operator , and switching to spherical coordinates, we have
We can now write the Hamiltonian for the zero angular momentum case
With application of this Hamiltonian to the eigenfunction we have
In particular for we have
Collecting all the results we have
A particle in a uniform field . Show that the expectation value of the position operator satisfies
This follows from Ehrehfest’s theorem once we formulate the force , in terms of a potential . That potential is
The Hamiltonian is therefore
Ehrehfest’s theorem gives us
Putting all the last bits together, and summing over the directions we have
For Hamiltonian eigenstates , , , obtain the matrix element in terms of the matrix element of .
I was able to get most of what was asked for here, with a small exception. I started with the matrix element for , which is
Next, computing the matrix element for we have
Except for the part of this expression, the problem as stated is complete. The relationship 2.25 is no help for with , so I see no choice but to leave that small part of the expansion in terms of .
Operator has eigenstates , with a unitary change of basis operation . Determine in terms of , and the operator and its eigenvalues for which are eigenstates.
Consider for motivation the matrix element of in terms of . We will also let . We then have
We also have
So it appears that the operator has the orthonormality relation required. In terms of action on the basis , let’s see how it behaves. We have
So we see that the operators and have common eigenvalues.
With , and , show that
We also have
Or, for ,
Problem 11. commutator of angular momentum with Hamiltonian.
Show that , where .
This follows by considering , and . Let
We now need to consider the commutators of the operators with and .
Let’s start with . In particular
So our commutator with is
Since , all terms cancel out, and the problem is reduced to showing that
Now assume that has a series representation
We’d like to consider the action of on this function
Thus as expected, implying .
 BR Desai. Quantum mechanics with basic field theory. Cambridge University Press, 2009.