# Problems in Quantum Mechanics

1. The wavelength spectrum of the radiation energy emitted from a system in thermal equilibrium is observes to have a maximum value which decreases with increasing temperature. Outline briefly the significance of this observation for quantum physics.

2. The “stopping potential” in a photoelectric cell depends only on the frequency v of the incident electromagnetic radiation and not on its intensity. Explain how the assumption that each photoelectron is emitted following the absorption of a single quantum of energy hv is consistent with this observation.

3. Write down the de Broglie equations relating the momentum and energy of free particle to, respectively, the wave number k and angular frequency w of the wave-function which describes the particle.

4. Write down the Heisenberg uncertainty Principle as it applies to the position x and momentum p of a particle moving in one dimension.

5. Estimate the minimum range of the momentum of a quark confined inside a proton size 10 ^ -15 m.

6. Explain briefly how the concept of wave-particle duality and the introduction of a wave packet for a particle satisfies the Uncertainty Principle.

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#### Solution Summary

We solve several problems in quantum mechanics.

# problems in Quantum Mechanics

1. show that the commutator obeys:

[A,B] = -[B,A]

[A,B+C]=[A,B] + [A,C]

[A,BC]=[A,B]C+B[A,C]

[A,[B,C]]+[B,[C,A]]+[C,[A,B]]=0

Given the fundamental commutator relation between momentum and position [x,p] = ih

show that:

a. [x^n,p] = ihn*x^(n-1)

b. [x,p^n] = ihn*p^(n-1)

c. show that if f(x) can be expanded in polynomial in x and g(p) can be expanded in a polynomial in p, then [f(x),p] = ih*df/dx

and [x,g(p)] = ih * dg/dp

for problem 3 that deals with Hamiltonian, observables, diagonalization and energy values see attached file

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