The physical derivation of annihilation operator?

[Pring]

From P. Meystre's book elements of quantum optics (Many labels of equations are wrong) Page 83, the annihilation operator and creation operator, which are helpful to discuss harmonic oscillator, are defined as

$a=\frac{1}{\sqrt{2\hbar\Omega}}(\Omega q+ip),\\ a^\dagger=\frac{1}{\sqrt{2\hbar\Omega}}(\Omega q-ip).$

This definition can't satisfy anyone's physical curiosity. How we deduce these operators more naturally?

 

[DrClaude]

Have a look at section 4.1 in http://www.ks.uiuc.edu/Services/Class/PHYS480/qm_PDF/chp4.pdf

 

[Pring]

Have a look at section 4.1 in http://www.ks.uiuc.edu/Services/Class/PHYS480/qm_PDF/chp4.pdf

It is just a definition.

 

[DrClaude]

It is just a definition.

I don't understand what you mean. The document a linked to shows how one can posit which form operators $\hat{a}$ and $\hat{a}^+$ should have, and why they are ladder operators.

That's how I interpreted your original question

How we deduce these operators more naturally?

with en emphasis on deduce.

If this is not what you are looking for, please give a more detailed question.

 

[Pring]

I don't understand what you mean. The document a linked to shows how one can posit which form operators $\hat{a}$ and $\hat{a}^+$ should have, and why they are ladder operators.

That's how I interpreted your original question

with en emphasis on deduce.

If this is not what you are looking for, please give a more detailed question.

Thank you. I want to deduce the ladder operators based on physical reasons, not a mathematical skill and definition like this:

 

[DrClaude]

Thank you. I want to deduce the ladder operators based on physical reasons, not a mathematical skill and definition like this:

I'm not aware of a more "physical" way of finding out what these operators are.

You know, it is often like that in physics. Clever people play around with the maths, and figure out things that were unexpected. I don't know if some ever sat down with the intention of figuring out "What is the operator that will take any eigenfunction of the harmonic oscillator and give back the next higher eigenfunction?", compared to someone just realizing, "Hey, look at what this operator I found by playing around with the algebra does!"

 

[Pring]

I'm not aware of a more "physical" way of finding out what these operators are.

You know, it is often like that in physics. Clever people play around with the maths, and figure out things that were unexpected. I don't know if some ever sat down with the intention of figuring out "What is the operator that will take any eigenfunction of the harmonic oscillator and give back the next higher eigenfunction?", compared to someone just realizing, "Hey, look at what this operator I found by playing around with the algebra does!"

Yeah, many historical events prove your viewpoint, such as Dirac equation, positron. But more important thing is that we should understand it more deeply and find more simpliy deduction, rather than such mathematical skills.

 

[Chandra Prayaga]

There is an "explanation" given in the fantastic textbook on Quantum Mechanics by Griffiths. The Schrödinger equation is a 2nd order differential equation. For a harmonic oscillator, the hamiltonian H is quadratic in position and momentum. If we can write the hamiltonian as a product of two first order terms, then the original 2nd order equation can be written as two first order equations, which are easier to solve. So the attempt is to find first order factors of H. When you try that, (ref: Griffiths), you find that because of the commutation relation between x and p, there is a leftover term, and the best you can do is, H = (a^†a + 1/2)ħω, where a^† and a are the usual creation and annihilation operators. Once they are defined thus, the algebra follows.

 

[DrClaude]

There is an "explanation" given in the fantastic textbook on Quantum Mechanics by Griffiths. The Schrödinger equation is a 2nd order differential equation. For a harmonic oscillator, the hamiltonian H is quadratic in position and momentum. If we can write the hamiltonian as a product of two first order terms, then the original 2nd order equation can be written as two first order equations, which are easier to solve. So the attempt is to find first order factors of H. When you try that, (ref: Griffiths), you find that because of the commutation relation between x and p, there is a leftover term, and the best you can do is, H = (a^†a + 1/2)ħω, where a^† and a are the usual creation and annihilation operators. Once they are defined thus, the algebra follows.

It is the same approach that was in the link I posted.

 

[Chandra Prayaga]

It is the same approach that was in the link I posted.

Absolutely. Thought I would put it into words and add the reference of Griffiths.

 

[vanhees71]

I think the introduction of these operators is very natural given the large symmetry of the harmonic-oscillator Hamilton, which goes beyond the obvious space-time symmetries. It has a socalled dynamical symmetry. The homogeneous $N$-dimensional oscillator has a $\mathrm{SU}(N)$ symmetry.

For $N=1$ this can be seen as follows. You start with the Hamiltonian (e.g., from the heuristics of "canonical quantization" of the classical harmonic oscillator)
$$\hat{H}=\frac{\hat{p}^2}{2m}+\frac{m \omega^2}{2} \hat{x}^2.$$
Now you define
$$\sqrt{m \omega^2} \hat{x}^2=\hat{\xi}, \quad \frac{1}{\sqrt{2m}}=\hat{\Pi},$$
and then the Hamiltonian reads
$$\hat{H}=\frac{1}{2} (\hat{\Pi}^2+\hat{\xi}^2).$$
You immideately see that this is symmetric under $\mathrm{O}(2)$ rotations
$$\begin{pmatrix} \hat{\xi}' \\ \hat{\Pi}' \end{pmatrix} = \begin{pmatrix} \cos \varphi & \sin \varphi \\ -\sin \varphi & \cos \varphi \end{pmatrix} \begin{pmatrix} \hat{\xi} \\ \hat{\Pi} \end{pmatrix}.$$
Now it is much simpler to deal with the combination
$$\hat{a}=\sqrt{\frac{1}{2 \hbar \omega}}(\hat{\xi} +\mathrm{i} \hat{\Pi}),$$
because then the above $\mathrm{O}(2)$ transformation can be written as $\mathrm{U}(1)$ transformation
$$\hat{a}'=\exp(-\mathrm{i} \varphi) \hat{a}.$$
The Hamiltonian must thus be a function of $\hat{a}^{\dagger} \hat{a}$ and indeed you find, using the Heisenberg commutators for $\hat{x}$ and $\hat{p}$
$$\hat{H}=\hbar \omega \left (\hat{a}^{\dagger} \hat{a} + \frac{1}{2} \right ).$$
You can solve the complete eigenvalue problem for $\hat{H}$ by analyzing the algebra in terms of $\hat{a}$ and $\hat{a}^{\dagger}$, which generate the harmonic-oscillator energy eigenstates in terms of a Fock space, because $\hat{a}$ turns out to be a ladder operator, a socalled quasiparticle single-mode annihilation operator. The quasiparticles in this case can be called the most simple example for phonons.

Indeed you can quantize a solid crystal in the most simple model as a set of harmonic oscillators, describing the crystal-lattice vibrations has harmonic oscillations around the equilibrium positions of the atoms/molecules making up the solid. That's then equivalent to a free gas of bosonic quasiparticles, which are called phonons, because the lattice vibrations are nothing else than sound waves.

 

[atyy]

Just to give a handwavy version of what vanhees71 said: it comes about naturally if one considers the "Fourier" domain, that solutions to the free wave equation can be considered superpositions of sinusoids. These are the sound waves that vanhees71 refers to.

 

[A. Neumaier]

If you want to understand the origin of annihilation and creation operators from a more general perspective, you should read

F Cooper, A Khare, U Sukhatme, Supersymmetry and quantum mechanics, Physics Reports, 1995

 

[stevendaryl]

Here's another perspective on the harmonic oscillator.

Do you know about the Heisenberg picture? In the usual Schrödinger picture, p and x are time-independent operators, and the wave function \psi is a function of time. In the Heisenberg picture, it's the other way around: p and x are time-dependent. One nice thing about the Heisenberg picture is that the Heisenberg equations of motion look almost like the classical ones, but with some quantum "correction terms". These correction terms turn out to vanish for the harmonic oscillator.

So let's work in the Heisenberg picture. Since classically we know that the harmonic oscillator results in periodic motion with angular frequency \omega = \sqrt{\frac{k}{m}}, we can guess that quantum-mechanically, p and x are similarly periodic:

x = A e^{-i \omega t} + A^{\dagger} e^{+i \omega t}

That's a perfectly general form for a Hermitian operator with angular frequency \omega.

p = m \dot{x} = -i m \omega (A e^{-i \omega t} - A^\dagger e^{+i \omega t})

The commutation relation for p and x gives us:

-i m \omega ([A, A^\dagger] - [A^\dagger, A]) = -i \hbar

or [A, A^\dagger] = \frac{\hbar}{2m \omega}

In terms of A and A^\dagger,

E = \frac{m \omega^2}{2} x^2 + \frac{1}{2m} p^2 = m \omega^2(A A^\dagger + A^\dagger A)= m \omega^2 (2 A^\dagger A + [A, A^\dagger]) = 2 m \omega^2 A^\dagger A + \frac{\hbar \omega}{2}

The usual raising and lowering operators are:

a = \sqrt{\frac{2m \omega}{\hbar}} A
a^\dagger = \sqrt{\frac{2m \omega}{\hbar}} A^\dagger