Applied Quantum Mechanics: for Engineers and Physicists 1st. Edition by A. F. J. Levi | 9780521520867 | 052152086X | Holooly

Quantum Mechanics

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. ...

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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 ...

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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 = ...

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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? ...

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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 . ...

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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 ...

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(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 ...

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(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? ...

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A two-dimensional harmonic oscillator with motion ...

Question: 2.9

In Section 2.2.3.2 it was stated that the degeneracy of state ψnlm in a hydrogen atom is n². Show that this is so by proving ∑l=0^l=n-1(2l+1) = n² ...

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In Section 2.2.3.2 the energy of the hydrogen atom...

Question: 10.2

Calculate the energy levels of an anharmonic oscillator with potential of the form V(x) = κ/2 x² + ξ x³ ћω where κ is the spring constant for a harmonic potential. Show that the difference between two adjacent perturbed levels is En − En−1 = ћ ω (1 − 15ξ² (ћ/ mω)³ n/2). A heterodi-atomic molecule ...

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The Hamiltonian for the one-dimensional harmonic o...

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