All parts of this problem refer to the harmonic oscillator. A) Use the expressions for the operators a+ and a− that we saw in class and the fact that xˆ and pˆ are Hermitian operators to show that a+ is the Hermitian conjugate of a− and vice versa. B) Express the operators xˆ and pˆ as a combination of the operators a+ and a−. C) Based on the above, show that the expected value of both xˆ and pˆ is zero at any steady state of the oscillator.

All parts of this problem refer to the harmonic oscillator. A) Use the expressions for the operators a+ and a− that we saw in class and the fact that xˆ and pˆ are Hermitian operators to show that a+ is the Hermitian conjugate of a− and vice versa. B) Express the operators xˆ and pˆ as a combination of the operators a+ and a−. C) Based on the above, show that the expected value of both xˆ and pˆ is zero at any steady state of the oscillator.