PHYS3111 – Quantum Mechanics Term 2, 2022
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PHYS3111 – Quantum Mechanics
Term 2, 2022
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Sometimes one can ﬁnd out useful things about a system without actually solv- ing the Schr¨odinger equation and ﬁnding wavefunctions. The virial theorem is one tool that can be used. Originally formulated for classical physics (thermo- dynamics) it relates the kinetic energy to the potential energy and hence the total energy of a system. For a single particle in a potential (in three dimen- sions), the quantum virial theorem reads
≈T( = ≈P . aV( (1)
where ≈T( is the expectation value of the kinetic energy operator = m〇2 and V is the potential.
(a) Consider the Coulomb potential
Using the virial theorem, determine the ratio of the average kinetic energy ≈T( to the total energy E of a particle in the Coulomb well.
(b) Consider a more general potential
V (γ) = 6γn
where 6 is some arbitrary constant and n is an integer. Using the virial theorem determine
i. the ratio of average kinetic energy to the total energy, ≈T(/E, and
ii. the ratio of average potential energy to the total energy, ≈V(/E .
iii. Apply this to the Harmonic oscillator and determine the ratio ≈T(/≈V(.
In this question we will apply the variational theorem to obtain an approxi- mate solution to the ground state energy of a particle in the one-dimensional triangular well
V (x) = )x) (2)
The exact solution of the Schr¨odinger equation in this case can be found with really horrible Airy functions. We are not going to do that.
(a) Instead, use the Variational Method as done in class to ﬁnd the ground state energy. I want you to consider any reasonable trial function you like, parameterised in as many variables a; b; ::: as you like. Take
≈ )Hˆ ) (
≈ ) (
and ﬁnd the minimum energy with respect to your parameters
A better trial function will lead to a lower energy. Find the lowest energy that you can.
(b) Repeat the exercise to approximate the energy of the ﬁrst excited state in the same potential.
Variational principle vs. perturbation theory
Consider a system in three-dimensions which can be approximated using only two bound states:
) o ( = Ro (r) Yoo (; ) (5) ) lm( = Rl (r) Ylm(; ) (6)
These have energies o and l , and orbital angular momentum l = 0 and l = 1, respectively. The upper state, ) lm(, is triply-degenerate.
A weak electric ﬁeld is applied in the z-direction, resulting in the perturbation
V = E z = E r cos (7)
(a) Show that the ﬁrst-order energy shift for ) o ( and ) lm( is zero. (b) Write an expression for the matrix element
W = ≈ o ) V ) lo ( (8)
Leave your answer in terms of integrals over the radial functions Ro and Rl , but evaluate the angular part.
(c) Write an expression for the second-order perturbation theory correction to the energies of the states ) o ( and ) lo ( in terms of o and l and W .
(d) The Hamiltonian of the unperturbed system (for m = 0) can be represented as a two-by-two matrix
Hˆo = /0(∈ o) 、
while the wave functions can be represented as the vectors
)wo ( = /0(1) 、 Y )wlo ( = /1(0) 、
so that Hˆo )wi ( = ∈i )wi ( for 乞 = 0Y 1.
The full Hamiltonian in this basis is
Hˆ = Hˆo + iV = /W(∈o) 、 (9)
since the perturbation iV does not mix states with diﬀerent m. Find the energies of the system (eigenvalues of the Hamiltonian).
(e) Show that for small electric ﬁelds E, and hence small W , the exact eigen- values correspond to the results of second-order perturbation theory. Hint: ^1 + α ● 1 + α/2 for small α .