Question

18. Nine identical spin- 1 2 particles are put into a 3D Harmonic Oscillator with classical frequency ω. What is the ground state energy for the 9 particles? Show your reasoning.

Answer #1

Consider the three-dimensional harmonic
oscillator.
Indicate the energy of the base state if twelve
identical particles (spin 1) are placed in the system that do not
interact with each other.

Description: I am studying the chapter of Identical Particles
for QM. My instructor only shown me how to deal with square wells,
I was wondering about the Harmonic Oscillator. Is it still the
similar treatment like Infinite Square Wells? Please help me solve
the problem I create.
I am new on this chapter so please SHOW ALL YOUR STEPS and be as
concise as possible !
Problem:
Three particles are confined in a 1-D harmonic
oscillator potential. Determine the energy...

A quantum mechanical simple harmonic oscillator has a 1st
excited state with energy 3.3 eV and there are eight spin-1/2
particles in the oscillator. How much energy is needed to add a
ninth electron. Explain and show your work

A quantum mechanical simple harmonic oscillator has a 1st
excited state with energy 3.3 eV and there are eight spin-1/2
particles in the oscillator. How much energy is needed to add a
ninth electron. Explain and show your work

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 quantum mechanical simple harmonic oscillator has a 1st
excited state with energy 3.3 eV and there are eight spin 1⁄2
particles in the oscillator. How much energy is needed to add a
ninth electron. Explain and show your work.

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...

To generate the excited states for the quantum
harmonic oscillator, one repeatedly applies
the raising operator ˆa+ to the ground state, increasing the energy
by ~ω with each step:
ψn = An(ˆa+)
nψ0(x) with En = (n +
1
2
)~ω
where An is the normalization constant and
aˆ± ≡
1
√
2~mω
(∓ipˆ+ mωxˆ).
Given that the normalized ground state wave function is
ψ0(x) = mω
π~
1/4
e
− mω
2~
x
2
,
show that the first...

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...

A system of 6 identical 1-dimensional harmonic oscillators (it
could be two atoms of an Einstein model solid). What is the
probability that the first oscillator has 2 quanta of vibrational
energy when the system has 3 quanta of vibrational energy?
Hint: You may find it useful to consider how many microstates
the system has and how many microstates the rest of the system has
when the first oscillator has 2 quanta.
Answer is 5/56
For the system considered in...

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