Expectation for the Harmonic Oscillator ( using dirac)

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• Somali_Physicist
In summary: You wrote a couple of things that don't make sense.In summary, the author is asking if there is an expected value for all odd energy levels, and is bit iffy about a step in their proof.

Somali_Physicist

I've been trying to form a proof using , using majorly dirac notation.There has been claims that its much better to use in QM.

The question i wanted to generally show that the expected value is Zero for all odd energy levels.I believe i have solved the question but I am a bit Iffy about a step i took:

<x>n = <Ψn|x|Ψn> = L
for a given Ψn = (A+)n(n!)-2
Energy eigen functions have definite parity, assume for all odd n's if one is zero the rest should follow.
Take n = 1
=> L = <(A+)(n!)-2|x|(A+)(n!)-2>
= (n!)-1 <(A+)|x|(A+)>
B = <(A+)|x|(A+)>
Define A+ = Lx + iC : B,C are Real
=> <Lx+iC|x|Lx+iC>
(Bit iffy after these steps)
= <Lx|x|Lx|> + <iC|x|iC>
= <L|x3|L>+<C|x|C>
as ∫x2n+1dx for limits [-∞,∞] and n =0,1,2,3...
=> 0
we find B=0
therefore
<x>n = 0
For odd ns.

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What about even values of ##n##? You are going about it backwards. Instead of replacing ##\psi_n##, write operator ##x## in terms of ##a^{\dagger}## and ##a## and use your knowledge of what ##a^{\dagger}|\psi_n>## and ##a|\psi_n>## are equal to.

kuruman said:
What about even values of ##n##? You are going about it backwards. Instead of replacing ##\psi_n##, write operator ##x## in terms of ##a^{\dagger}## and ##a## and use your knowledge of what ##a^{\dagger}|\psi_n>## and ##a|\psi_n>## are equal to.
Surely you wouldn't get an actual value for even values.That would be extremely counter intuitive, that said I will try your advice.

You wrote a couple of things that don't make sense.
Somali_Physicist said:
Ψn = (A+)n(n!)-2
This should be
$$|\psi_n\rangle = \frac{(A^\dagger)^n}{\sqrt{n!}} | 0 \rangle$$

Somali_Physicist said:
=> L = <(A+)(n!)-2|x|(A+)(n!)-2>
The notation here doesn't work. You can't have an operator in a bra or a ket. You should end up with something like
$$\langle 0 | A x A^\dagger |0 \rangle$$
and so on.

Somali_Physicist
Somali_Physicist said:
Surely you wouldn't get an actual value for even values.That would be extremely counter intuitive, that said I will try your advice.
Why is it so counter intuitive? In your proof, which needs fixing as @DrClaude suggested, you have the integral ∫x2n+1dx where n is odd. What would happen to this integral if n were even? Say n = 2k?

Somali_Physicist said:
The question i wanted to generally show that the expected value is Zero for all odd energy levels.
What do you mean by "energy", i.e. what is the Hamiltonian?

1. What is the expectation value for the position of a particle in the harmonic oscillator using Dirac notation?

The expectation value for the position of a particle in the harmonic oscillator using Dirac notation is given by x = <a+|X|a>, where a+ and a are the raising and lowering operators respectively.

2. How is the expectation value for momentum calculated in the harmonic oscillator using Dirac notation?

To calculate the expectation value for momentum using Dirac notation, we use the formula p = <a+|P|a>, where a+ and a are the raising and lowering operators and P is the momentum operator.

3. What is the relationship between the expectation values for position and momentum in the harmonic oscillator?

In the harmonic oscillator, the expectation value for position and momentum are related by the Heisenberg uncertainty principle. This means that as the uncertainty in position decreases, the uncertainty in momentum increases, and vice versa.

4. How is the expectation value for energy calculated in the harmonic oscillator using Dirac notation?

The expectation value for energy in the harmonic oscillator is given by E = <a+|H|a>, where H is the Hamiltonian operator.

5. How do we interpret the expectation value for energy in the harmonic oscillator?

The expectation value for energy in the harmonic oscillator represents the average energy of the system over time. It is a useful tool in understanding the behavior of the system and predicting its future state.