Question

5. The wavefunction of a particle is ?(?) = ??−??/2 for x>0 and ?(?) = ????/2 for x<0. Find the corresponding potential energy, constant A, and energy eigenvalue.

6. The hydrogen molecule H2 can be treated as a vibrating system
(simple harmonic oscillator), with an effective force constant ? =
3.5 × 10^3 eV/nm2. Compute the zero-point (ground state) energy
of

one of the protons in H2. How does it compare with the molecular
binding energy of 4.5 eV? Compute the amplitude of the zero-point
motion and compare with the atomic spacing of 0.074 nm.

Answer #1

. The hydrogen molecule H2 can be treated as a vibrating system
(simple harmonic oscillator), with an effective force constant ? =
3.5 × 10^3 eV/nm2. Compute the zero-point (ground state) energy of
one of the protons in H2. How does it compare with the molecular
binding energy of 4.5 eV? Compute the amplitude of the zero-point
motion and compare with the atomic spacing of 0.074 nm

In Classical Physics, the typical simple harmonic oscillator is
a mass attached to a spring. The natural frequency of vibration
(radians per second) for a simple harmonic oscillator is given by
ω=√k/m and it can vibrate with ANY possible energy whatsoever.
Consider a mass of 135 grams attached to a spring with a spring
constant of k = 1 N/m. What is the Natural Frequency (in rad/s) of
vibration for this oscillator?
In Quantum Mechanics, the energy levels of a...

A harmonic oscillator with the usual PE of V(x)= (.5)kx^2
perturbed by a small change to the spring constant k -->(1+E)k,
with E<<1.
1. Write the new energy eigenvalues, making sure any parameters
are clearly defined.
2. Expand the eigenvalue expression in a power series in E up to
the second order using a Taylor series expansion.
3. What is the perturbation Hamiltonian in the problem?

Consider the wave function at t = 0, ψ(x, 0) = C sin(3πx/2)
cos(πx/2) on the interval 0 ≤ x ≤ 1.
(1) What is the normalization constant, C?
(2) Express ψ(x,0) as a linear combination of the eigenstates of
the infinite square well on the interval, 0 < x < 1. (You
will only need two terms.)
(3) The energies of the eigenstates are En =
h̄2π2n2/(2m) for a = 1. What is
ψ(x, t)?
(4) Compute the expectation...

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