Coherent States

General Concept

A coherent state is an eigenstate of the lowering operator for the quantum harmonic oscillator. It is the state which behaves most like the classical analog of the Simple Harmonic Oscillator. That is, its uncertainty relation is minimized and it oscillates back and forth like a mass on a spring.

Formulas

To derive the coherent state of the quantum harmonic oscillator we look for eigenstates of the lowering operator

$a|\varphi ⟩=\varphi |\varphi ⟩$

We know for any state we can write this as a sum of the basis states, so we may write

$|\varphi ⟩={\sum }_{n=0}^{\infty }{c}_n|n⟩$

$c_n=⟨\varphi |n⟩$

Now lets apply the lowering operator to this state

$a|\varphi ⟩={\sum }_{n=0}^{\infty }{c}_{n}a|n⟩={\sum }_{n=1}^{\infty }{c}_n\sqrt{n}|n-1⟩={\sum }_{n=0}^{\infty }{c}_{n+1}\sqrt{n+1}|n⟩$

Thus we have

$a|\varphi ⟩={\sum }_{n=0}^{\infty }{c}_{n+1}\sqrt{n+1}|n⟩=\varphi |\varphi ⟩=\varphi {\sum }_{n=0}^{\infty }{c}_n|n⟩$

$⟹{\sum }_{n=0}^{\infty }[c_{n+1}\sqrt{n+1}-\varphi c_n]|n⟩=0$

Which can only be true if for every n $c_{n+1}\sqrt{n+1}-\varphi c_n=0$ thus we have a recursion relation

$c_{n+1}=\frac{\varphi c_n}{\sqrt{n+1}}$

Now lets write the first couple coefficients

$c_1=\frac{\varphi c_0}{\sqrt{1}},c_2=\frac{\varphi c_1}{\sqrt{2}}=\frac{{\varphi }^2{c}_0}{\sqrt{1}\sqrt{2}},c_3=\frac{\varphi c_2}{\sqrt{3}}=\frac{{\varphi }^3{c}_0}{\sqrt{1\cdot 2\cdot 3}}$

So we may write this as

$c_n=\frac{{\varphi }^n{c}_0}{\sqrt{n!}}$

Finally we must normalize these coefficients

${\sum }_{n=0}^{\infty }\frac{|\varphi {|}^{2n}}{n!}|c_0{|}^2=1$

$|c_0{|}^2{\sum }_{n=0}^{\infty }\frac{|\varphi {|}^{2n}}{n!}=1$

But notice that inside the sum is just the Taylor series for a gaussian, so

$|c_0{|}^2{e}^{|\varphi {|}^2}=1$

$⟹|c_0{|}^2=e^{-|\varphi {|}^2}$

We finally have then that

$c_0=e^{-\frac{|\varphi {|}^2}{2}}$

So we can write the coherent state as

$|\varphi ⟩=e^{-\frac{|\varphi {|}^2}{2}}{\sum }_{n=0}^{\infty }\frac{{\varphi }^n}{\sqrt{n!}}|n⟩$