Quantum harmonic oscillator, uncertainty relation

In summary, the position and momentum operators of a particle in simple harmonic motion can be expressed in terms of the annihilation and creation operators. These operators satisfy certain commutation relations and the eigenvalue equation can be written in bra notation. In a coherent state, the expectation values of the operators can be calculated. However, during the calculation of the uncertainty relation, there may be an error that leads to a final value of 0.
  • #1
phys-student
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0

Homework Statement


Consider a particle with mass m oscillates in a simple harmonic potential with frequency ω. The position, x, and momentum operator, p, of the particle can be expressed in terms of the annihilation and creation operator (a and a respectively):
x = (ħ/2mω)^0.5 * (a + a)
p = i(ħmω/2)^0.5 * (a - a)
The annihilation and creation operators satisfy the following commutation relations:
[a,a] = 1
The eigenvalue equation of a is found as:
a|α> = α|α>
Where |α> is a coherent state
a) Find the expectation value <a> in coherent state |α>
b) Find the expectation values of <aa>, <aa>, and <aa>
c) Calculate the uncertainty relation (<(x-<x>)2><(p-<p>)2>)0.5

Homework Equations


Relevant equations given above

The Attempt at a Solution


I've worked through most of the problem and only noticed that I may have done something wrong when I got to part c, where the expression that I get for <x> causes the expectation value <(x-<x>)2> to be equal to 0, which also means the uncertainty relation is equal to 0. I think the error may be when I wrote the eigenvalue equation for a in bra notation like this:
<α|a = <α|α*
Is this the correct way of expressing the eigenvalue equation given in the question in bra notation?
 
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  • #2
phys-student said:
<α|a† = <α|α*
Is this the correct way of expressing the eigenvalue equation given in the question in bra notation?
That's correct. But I don't know where you went wrong unless you provide your calculation.
 
  • #3
If that is correct then the expectation values are as follows; <a>=α*, <a> = α, <aa> = α2, <aa> = α*2, and <aa> = <aa> = αα*.

This means that the expectation value <x> is: <x> = (ħ/2mω)0.5(<a> + <a>) = (ħ/2mω)0.5(α* + α)

So the expression (x-<x>)2 = x2 - 2x(ħ/2mω)0.5(α* + α) + (ħ/2mω)(α* + α)2

So the expectation value from the uncertainty relation is: <(x-<x>)2> = <x2> - 2*(ħ/2mω)0.5(α*+α)<x> + (ħ/2mω)(α* + α)2

The expectation value <x2> is:
<x2> = (ħ/2mω)(<aa> + <aa> + <aa> + <aa>) = (ħ/2mω)(α* + α)2

If you sub the expression for <x> and <x2> back into the expression for <(x-<x>)2> it ends up reducing to 0
 
  • #4
phys-student said:
<aa†> = <a†a> = αα*.
That's where you go wrong. Your ##\langle a^\dagger a \rangle## is correct, but your ##\langle aa^\dagger \rangle## is not. To calculate the latter, you could have used the commutation between the annihilation and raising operators and ##\langle a^\dagger a \rangle##.
 
  • #5
Okay so you are saying that <aa> = αα* is correct, then using the commutation relation:

[a,a] = aa - aa = 1, therefore: aa = 1 + aa

So then the expectation value of aa is: <aa> = 1 + <aa> = 1 + αα*

Is that correct?
 
  • #6
phys-student said:
Is that correct?
Yes.
 
  • #7
Okay thanks
 

1. What is a quantum harmonic oscillator?

A quantum harmonic oscillator is a theoretical system that describes the behavior of a particle trapped in a potential well that varies harmonically with time. It is a fundamental model in quantum mechanics and can be used to understand the behavior of atoms, molecules, and other physical systems.

2. How does the uncertainty relation apply to the quantum harmonic oscillator?

The uncertainty relation, also known as Heisenberg's uncertainty principle, states that it is impossible to simultaneously know the exact position and momentum of a particle. In the case of the quantum harmonic oscillator, this means that the more precisely we know the position of the particle, the less precisely we can know its momentum, and vice versa.

3. What is the significance of zero-point energy in the quantum harmonic oscillator?

The quantum harmonic oscillator has a minimum energy state, known as the ground state, which is not zero but rather a finite value called the zero-point energy. This energy is a consequence of the uncertainty principle and is the lowest energy state that a quantum system can have.

4. How does the energy level spacing in the quantum harmonic oscillator differ from classical harmonic oscillators?

In classical harmonic oscillators, the energy levels are evenly spaced, while in quantum harmonic oscillators, the energy levels are not evenly spaced. This is due to the quantization of energy in quantum mechanics, where energy can only exist in discrete amounts rather than being continuous.

5. What are some real-world applications of the quantum harmonic oscillator?

The quantum harmonic oscillator has various applications in fields such as quantum computing, spectroscopy, and quantum optics. It is also used to model the behavior of electrons in atoms and the vibrations of molecules. Additionally, the principles of the quantum harmonic oscillator are essential in understanding the properties of materials and their behavior at the atomic level.

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