Harmonic Oscillator, Ladder Operators, and Dirac notation

In summary, the conversation discusses the definition of the state | \alpha > as an eigenstate of the lowering operator \hat{a}, and the use of Dirac notation to represent this statement. The conversation also briefly touches on the steps one would take to prove this definition and the potential difficulty of the calculation.
  • #1
MaximumTaco
45
0
Defining the state [itex]| \alpha > [/itex] such that:
[tex]| \alpha > = Ce^{\alpha {\hat{a}}^{\dagger}} | 0 >\ ,\ C \in \mathbf{R};\ \alpha \in \mathbf{C};[/tex]

Now, [itex]| \alpha >[/itex] is an eigenstate of the lowering operator [itex]\hat{a}[/itex], isn't it?

In other words, the statement that [itex] \hat{a} | \alpha >\ =\ \alpha | \alpha > [/itex] is true, right?
(for the eigenvalue [itex]\alpha[/itex]?)

Is that the correct use of the Dirac notation?

How might one go about proving the above?
 
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  • #2
Anybody? I'd really appreciate some advice here.

Cheers.
 
  • #3
I got the same result. Was there step where you had

[tex]
a (a^\dagger)^k = k (a^\dagger)^{k-1} + (a^\dagger)^k a
[/tex]

It looks like it's an eigenstate of lowering operation then. Pathological state...

I'm not sure about what is precisly correct notation, looked fine to me, but at least the latex symbols \langle and \rangle look better than < and > :wink:

Adding with edit:

Hups, I didn't notice your last question. I thougth you had calculated that, and were wondering how such strange state could exist. Steps to prove it are these. Use series expansion of the exponential. Prove the commutation rule of [tex]a[/tex] and [tex](a^\dagger)^k[/tex] with induction (or with some other technique). Calculate.
 
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  • #4
I wouldn't call coherent states "pathological states".

The way you would go about proving it would probably be by construction. Start by declaring a state [tex]|\alpha \rangle[/tex] to be an eigenstate of the lowering operator, then expand that state in the basis of the harmonic oscillator and see what comes out. Then you can equate it to the exponential of the raising operator by looking at what comes out and saying "AH HA!".

That's probably the least algebraically intensive method of solving the problem, unless you really like wrestling with commutators.
 
  • #5
I didn't know anything about coherent states, but the calculation wasn't too difficult so I replied. I'll take the pathological comment back.
 

1. What is a harmonic oscillator?

A harmonic oscillator is a physical system that exhibits periodic motion around a stable equilibrium point. It can be described mathematically using the harmonic oscillator equation, which is a second-order linear differential equation.

2. What are ladder operators in relation to the harmonic oscillator?

Ladder operators are mathematical operators that are used to describe the energy levels and transitions of a quantum harmonic oscillator. They allow us to understand the quantization of energy in the system and how the system moves between energy levels.

3. How do ladder operators relate to Dirac notation?

Dirac notation is a mathematical notation used in quantum mechanics to represent states and operators. Ladder operators are commonly written in Dirac notation, with the creation operator represented as a dagger symbol and the annihilation operator represented as a regular symbol.

4. How do ladder operators affect the energy levels of a harmonic oscillator?

Ladder operators act on the energy states of a harmonic oscillator by either increasing or decreasing the energy of the system. The creation operator increases the energy by one quantum, while the annihilation operator decreases the energy by one quantum.

5. Can ladder operators be used for other systems besides the harmonic oscillator?

Yes, ladder operators have applications in other quantum systems besides the harmonic oscillator. They are commonly used in the study of angular momentum and spin, as well as in field theory and the creation and annihilation of particles in quantum field theory.

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