Question

For a particle in the harmonic oscillator potential, calculate the followings in any stationary state n:

a) 〈?̂?̂〉? b) 〈?̂?̂〉? c) 〈[?̂, ?̂]〉n

Answer #1

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...

A particle in a simple harmonic oscillator potential V (x) = 1
/2mω^2x^2 has an initial wave function
Ψ(x,0) = (1/ √10)(3ψ1(x) + ψ2(x)) ,
where ψ1 and ψ2 are the stationary state solutions of the ﬁrst and
second energy level. Using raising and lowering operators (no
explicit integrals except for orthonomality integrals!) ﬁnd
<x> and <p> at t = 0

Calculate the standard deviation of momentum for the second
excited state of a harmonic oscillator.

Consider an electron bound in a three dimensional simple
harmonic oscillator potential in the n=1 state. Recall that the
e- has spin 1/2 and that the n=1 level of the oscillator
has l =1. Thus, there are six states {|n=1, l=1, ml,
ms} with ml= +1, 0, -1 and ms =
+/- 1/2.
- Using these states as a basis find the six states with
definite j and mj where J = L +s
- What are the energy levels...

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?

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?

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?

Calculate the expectation value of momentum for the second
excited state of the harmonic oscillator. Show steps for both
integral and bracket notation please.

Q. 2(a) For the first excited state of harmonic oscillator,
calculate <x>, <p>, <x2>, and <p2>
explicitly by spatial integration. (b) Check that the
uncertainty principle is obeyed.
(c) Compute <T> and <V>. .

Consider a 17O2 molecule as a simple harmonic oscillator.
a. Calculate the reduced mass of this molecule (in units of
kg).
b. If the force constant is 1750 N/m, what is the vibrational
frequency (in cm-1)? c. Calculate the frequency shift compared to
18O2 (in cm-1)?

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