Question

5)
Unlike the particle in the 1D box and the harmonic oscillator, the
energy of the ground state of the 2D rigid rotor is zero. What is
the difference between these cases that allows the energy to be
zero for the rigid rotor in 2D?

Answer #1

which system does not have a zero point energy? a. particle in
one dimensional box (b). one dimensional harmonic oscillator. (c)
two particle rigid rotor d) hydrogen atom

a)
For a 1D linear harmonic oscillator find the first order
corrections to the ground state due to the Gaussian perturbation.
b) Find the first order corrections to the first excited
state.
Please show all work.

The ground-state energy of a harmonic oscillator is 6 eV.
If the oscillator undergoes a transition from its n=3 to n=2 level
by emitting a photon, what is the energy (in eV) of the emitted
photon?

For
1D particle-in-a-box, if a wavefunction corresponds to ground state
and 1st excited state: Is the wave function an eigenfunction?

The normalized wave functions for the particle is in a 1D box of
length L., with limits on x = 0 and x = L. V (x) = 0 for 0 <= x
<= L and V (x) = Infinity elsewhere. The probability of a
particle being between x = 0 and x = L / 8 in the ground quantum
state (n = 1) should be calculated.

Quantum mechanics:
Consider a particle initially in the ground state of the
one-dimensional simple harmonic oscillator. A uniform electric
field is abruptly turned on for a time t and then abruptly turned
off again. What is the probability of transition to the first
excited state?

For a particle in the first excited state of harmonic oscillator
potential,
a) Calculate 〈?〉1, 〈?〉1, 〈? 2〉1, 〈? 2〉1.
b) Calculate (∆?)1 and (∆?)1.
c) Check the uncertainty principle for this state.
d) Estimate the length of the interval about x=0 which
corresponds to the classically allowed domain for the first excited
state of harmonic oscillator.
e) Using the result of part (d), show that position uncertainty
you get in part (b) is comparable to the classical range of...

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 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

Consider a one-dimensional harmonic oscillator, in an energy
eigenstate initially (at t=t0), to which we apply a time
dependent force F(t).
Write the Heisenberg equations of motion for x and for p.
Now suppose F is a constant from time t0 to time
t0+τ(tau), and zero the rest of the time. Find the
average position of the oscillator <x(t)> as a function of
time, after the force is switched off.
Find the average amount of work done by the force,...

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