Quantum Mechanics Answers

(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?

3) (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 (x, t)|²dx = 0]

(b) Show that

d(p)

dt

=

dx

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

2) Consider a 2D infinite potential well with the potential U(x, y) = 0 for 0 ≤ x ≤ a & 0 ≤ y ≤ B, and U(x, y) = ∞, otherwise. Solve the time-independent Schrodinger equation, and find the normalized wavefunction and the corresponding energies.

Tritium 3H (mt = 3.01605u) has a half-life of 12.3 years and releases 0.0186 MeV of energy per decay. What is the reason energy is released for a 4.1 g sample of Tritium?

In the amusement park ride Mr. Freeze at Six Flags, riders are uniformly accelerated from rest by magnetic induction motors along a 70 meter horizontal track in just 5 seconds. While accelerating, friction and air drag exert 500N of force on the train. The train then coasts through the loops and turns for the remainder of the ride. A typical train loaded with passengers has a mass of 2500 kg.

The wave function Ψ(x, t) = A exp [i(k1x − ω1t)] + A exp [i(k2x − ω2t)] is a superposition of two free-

particle wave functions. Both k1 and k2 are positive.

(a) Show that this wave function satisfies the Schodinger equation for a free particle of mass m.

(b) Find and plot the probability distribution function for Ψ(x, 0).