- #1

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- Homework Statement
- Given the quantum-optics coherent states

$$|\alpha \rangle = \exp \Big(\frac{|\alpha|^2}{2}\Big) \sum_{n=0}^{\infty} \frac{\alpha^n}{\sqrt{n!}} |n \rangle$$

Show that

$$\langle (\Delta X)^2 \rangle_{\alpha} = \langle (\Delta P)^2 \rangle_{\alpha} = \frac 1 4$$

Where

##|n \rangle## are the photon number states

$$X=\frac{a+a^*}{2}$$

$$P=\frac{a-a^*}{2i}$$

$$a|\alpha \rangle = \alpha |\alpha \rangle$$

- Relevant Equations
- $$|\alpha \rangle = \exp \Big(\frac{|\alpha|^2}{2}\Big) \sum_{n=0}^{\infty} \frac{\alpha^n}{\sqrt{n!}} |n \rangle$$

I've tried to square and compare ##\Delta X## and ##\Delta P## but they are not equal

I have to say I am pretty lost here and a hint would be appreciated.

I have studied coherent states and I know how to proof some properties related to it.

For instance, I see how to proof that the state is normalized:

$$\langle \alpha|\alpha \rangle = \exp(-|\alpha|^2) \Big(\sum_{m=0}^{\infty} \langle m| \frac{\alpha^{*m}}{\sqrt{m!}}\Big) \Big(\sum_{n=0}^{\infty} \frac{\alpha^{n}}{\sqrt{n!}}|n\rangle \Big) $$

Based on ##\langle m | n \rangle = \delta_{mn}## we indeed get ##\langle \alpha | \alpha \rangle = 1##

More can be found in Problem 1.1 https://ia800108.us.archive.org/32/items/FranzMandlGrahamShawQuantumFieldTheoryWiley2010/Franz%20Mandl%2C%20Graham%20Shaw-Quantum%20Field%20Theory-Wiley%20%282010%29.pdf; I also attached the complete solution of it.

Thanks

I have to say I am pretty lost here and a hint would be appreciated.

I have studied coherent states and I know how to proof some properties related to it.

For instance, I see how to proof that the state is normalized:

$$\langle \alpha|\alpha \rangle = \exp(-|\alpha|^2) \Big(\sum_{m=0}^{\infty} \langle m| \frac{\alpha^{*m}}{\sqrt{m!}}\Big) \Big(\sum_{n=0}^{\infty} \frac{\alpha^{n}}{\sqrt{n!}}|n\rangle \Big) $$

Based on ##\langle m | n \rangle = \delta_{mn}## we indeed get ##\langle \alpha | \alpha \rangle = 1##

More can be found in Problem 1.1 https://ia800108.us.archive.org/32/items/FranzMandlGrahamShawQuantumFieldTheoryWiley2010/Franz%20Mandl%2C%20Graham%20Shaw-Quantum%20Field%20Theory-Wiley%20%282010%29.pdf; I also attached the complete solution of it.

Thanks