Quantum Mechanics Answers

What is the relation between the size of a Bohr atom and the size of a Schroedinger atom?

Waves on deep water with surface tension T and density 𝜌 are governed by the dispersion relation 𝜔2=𝑔𝑘+𝑇 𝜌𝑘3. Calculate the phase and group velocities of the waves. Find the wave number kc at which phase velocity reaches a minimum. What is the group velocity for this wave number?

With a neat diagram, explain the working of electron microscopes. Briefly explain the role of each experimental component. Compare the resolution of electron microscopes and optical microscopes. [Hint: Explanation of SEM & TEM is required]

A particle strikes a potential barrier of height U and width L. Derive an expression for the approximate transmission probability, if the energy of the particle E < U

A particle strikes a potential barrier of height U and width L. Derive an expression for the approximate transmission probability, if the energy of the particle E < U

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