Confusion about initial states and coherent states

[valanna]

I've found online that the coherent state of the harmonic oscillator is
$$|\alpha \rangle = c \sum_{n=0}^\infty \frac{\alpha^n}{\sqrt{n!}} | n\rangle$$
where
$$|n\rangle = \frac{(a^\dagger)^n}{\sqrt{n!}} |0\rangle$$
and |0> is called the initial state.
I've some code where I need to have this initial state for j=4, so it should be a 9 by 1 vector right?
How is this initial state found?

 

[A. Neumaier]

t should be a 9 by 1 vector right?

Harmonic oscillators need an infinite dimensional Hilbert space, 9 dimensions do no suffice. The formula you wrote is for unpolarized light. In the case of polarization (where j makes sense) you need a Fock space over a 2-mode 1-particle Hilbert space, and the formula gets more complicated.

 

[valanna]

Harmonic oscillators need an infinite dimensional Hilbert space, 9 dimensions do no suffice. The formula you wrote is for unpolarized light. In the case of polarization (where j makes sense) you need a Fock space over a 2-mode 1-particle Hilbert space, and the formula gets more complicated.

So the |0> is in Hilbert space? If there were a coherent state that was in a space where j=4, is there a method to find |0>?

 

[A. Neumaier]

$|0\rangle$ is the ground state (no oscillation, e.g., no light) of a harmonic oscillator Hamiltonian, in the space $L^2(\Rz)$ (or the equivalent Fock space).

in a space where j=4

What do you mean by this? A 1-particle space with angular momentum 4 but position and momentum ignored? In this case, the appropriate coherent states are very different - you need angular momentum coherent states.

 

[valanna]

$|0\rangle$ is the ground state (no oscillation, e.g., no light) of a harmonic oscillator Hamiltonian, in the space $L^2(\Rz)$ (or the equivalent Fock space).

What do you mean by this? A 1-particle space with angular momentum 4 but position and momentum ignored? In this case, the appropriate coherent states are very different - you need angular momentum coherent states.

Thank you, Those are what I'm looking at, I suppose I made the mistake in thinking the coherent state for the harmonic oscillator was the same because its equation is a similar format to the one I'm looking at.
The main difference is that the state I need to find is represented by |j,j> but I'm having trouble finding how that state is actually found. I see it used or similar states used but no value or formula is ever given. Is it something trivial I'm just missing or does it have no actual value?
Thank you very much for your help

 

[vanhees71]

I've found online that the coherent state of the harmonic oscillator is
$$|\alpha \rangle = c \sum_{n=0}^\infty \frac{\alpha^n}{\sqrt{n!}} | n\rangle$$
where
$$|n\rangle = \frac{(a^\dagger)^n}{\sqrt{n!}} |0\rangle$$
and |0> is called the initial state.
I've some code where I need to have this initial state for j=4, so it should be a 9 by 1 vector right?
How is this initial state found?

$|0 \rangle$ is not an arbitrary initial state but the ground state of the harmonic oscillator. It's just defined by $\hat{a}|0 \rangle=0$. It's of course also a coherent state with $\alpha=0$.

The initial state of a quantum system can be anything. It's determined by the preparation of the system at the initial time.