Applied Quantum Mechanics: for Engineers and Physicists

86 SOLVED PROBLEMS

Question: 6.8

The ground-state wave function of a particle of mass m in a harmonic potential is ψ0 = A0e^−x²/4α² , where α² = ћ/2mω. Derive the uncertainties in position and momentum, and show that they satisfy the uncertainty relation. ...

For the operator Â (\Delta A)^{2}=\langle(\...
Question: 6.9

The ground state and the second excited state of a charged particle of mass m in a one-dimensional harmonic oscillator potential are both occupied. What is the expectation value of the particle position x as a function of time? What happens to the expectation value if the potential is subject to a ...

The ground state and the second excited state of a...
Question: 6.10

Using the method outlined in Section 3.4, write a computer program to solve the Schrödinger wave equation for the first four eigenvalues and eigenstates of an electron with effective mass m*e = 0.07 × m0 confined to a parabolic potential well in such a way that V(x) = ((x − L/2)²/(L/2)²) eV and L = ...

We would like to use the method outlined in Sectio...
Question: 7.1

Calculate kF in two dimensions for a GaAs quantum well with electron density n = 10¹² cm^−2. What is the de Broglie wavelength for an electron at the Fermi energy? ...

We wish to calculate $k_{\mathrm{F}}$...
Question: 7.2

In this chapter we showed that for temperatures kBT « EF the chemical potential for carriers in three dimensions may be approximated to second order in kBT as µ ∼ EF − π²/12 (kBT)²/EF Derive an expression for the chemical potential to fourth order in kBT . ...

We wish to derive an expression for the chemical p...
Question: 10.7

(a) An electron moves in a one-dimensional box of length X. Apply the periodic boundary condition φ(x) = φ(x + X) to find the electron eigenfunction and eigenvalues. (b) Now apply a weak periodic potential V(x) = V(x + L) to the system, where X = N L and N is a large positive integer. Using nondege ...

(a) Here, we are interested in an electron of mass...
Question: 10.8

(a) What is the effect of applying a uniform electric field on the energy spectrum of an atom? (b) If spin effects are neglected, the four states of the hydrogen atom with quantum number n = 2 have the same energy, E^0. Show that when an electric fielis applied to hydrogen atoms in these states, th ...

(a) We are asked to predict the effect that applyi...
Question: 10.6

In this chapter we solved for the first excited state of a two-dimensional harmonic oscillator subject to perturbation W = κ′ x y. How do the three-fold degenerate energy E = 3ћω and the four-fold degenerate energy E = 4ћω separate due to the same perturbation? ...

A two-dimensional harmonic oscillator with motion ...
Question: 2.9