Quantum Mechanics Answers

Let us consider a Hermitian operator 𝐴̂, with eigenvalues  "\ufeffa_1 = 1\/2" , "a_2 = 3\/2" and "a_3=5\/2" operating in a 1-Dimensional space.

(a) Can this operator be associated with a measurable quantity? Provide a brief justification for your answer ?

(b) Describe the meaning of degeneracy of the operator and demonstrate that all the eigen-states of 𝐴̂ are non-degenerate.

Consider an intrinsic semiconductor crystal at room temperature, where kBT is 0.025 eV. The probability of a state close to the valence-band edge being occupied by a hole is 1.0 x 10-5. Calculate the band gap.

Suppose, I have normalized the wave function at some point of time. The wave

function evolves with time according to time dependent Schrodinger equation. How do

I know that the wave function remains normalized after some time?

[Hint: Show that 𝑑

𝑑𝑡

∫−∞

∞

|𝜓(𝑥,𝑡)|

2𝑑𝑥 = 0]

Show that

𝑑⟨𝑝⟩

𝑑𝑡

= ⟨−

𝜕𝑉

𝜕𝑥⟩

i.e. expectation values follow Newton’s law.

Show that dp/dt =_dv/dx

(a) Suppose, I have normalized the wave function at some point of time. The wave

function evolves with time according to time dependent Schrodinger equation. How do

I know that the wave function remains normalized after some time?

[Hint: Show that 𝑑

𝑑𝑡

∫−∞

∞

|𝜓(𝑥,𝑡)|

2𝑑𝑥 = 0]

(b) Show that

𝑑⟨𝑝⟩

𝑑𝑡

= ⟨−

𝜕𝑉

𝜕𝑥⟩

i.e. expectation values follow Newton’s law.

For virtual particles:

(a) [2 points] clearly define the concept of virtual particle;

(b) [2 points] name an example of such a particle with the physical context or model

in which it occurs.

(c) Extra credit [1 point] what other system(s) can be described with such a model

(other than those from Particle Physics)?

For an electron in the l = 2 state:

(a) enumerate all the possible values of quantum numbers j and mj ;

(b) draw the corresponding vector diagrams;

(c) estimate the maximum value of the spin-orbit coupling energy E. 

A silver wire that is 2.5 mm in diameter and 37 cm long carries a current of 169 mA. How many electrons per second pass a given cross-section of the wire?

For laser action to occur, the medium used must have three energy levels. What must be the nature of these levels? Why is three the minimum number?