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1. Homework Statement

1. Homework Statement

I have to determine a linear combination of the states [itex]|0\rangle, |1\rangle[/itex], of a one dimensional harmonic oscillator, so that the expectation value [itex]\langle x \rangle [/itex] is a maximum.

## Homework Equations

[itex]x=\sqrt{\frac{\hbar}{2m\omega}}(a+a^\dagger)[/itex]

## The Attempt at a Solution

I set [itex]|\alpha \rangle\equiv c_1|1\rangle + c_2|0\rangle[/itex]

Then I calculate [itex] \langle \alpha | x | \alpha \rangle = \sqrt{\frac{\hbar}{2m\omega}} (c_1^*\langle1| + c_2^*\langle 0|)(a+a^\dagger)(c_1|1\rangle + c_2|0\rangle)=\sqrt{\frac{\hbar}{2m\omega}}[c_1^*(c_1+c_2)+c_2^*c_2] = \sqrt{\frac{\hbar}{2m\omega}}[1+c_1^*c_2][/itex]

I get the last equation because of the normalization condition:

[itex] c_1^*c_1+c_2^*c_2=1[/itex]

This is where I don't know how to continue. Is this even the right approach?

(I'm not allowed to use the wave functions.)

Anything is appreciated.

Kind regards

Alex