Consider the 1D harmonic oscillator. a) Find the normalized eigenstates of the annihilation operator a, denoted by |>, with eigenvalue in terms of the

Consider the 1D harmonic oscillator.
a) Find the normalized eigenstates of the annihilation operator a, denoted by |α>, with eigenvalue α in terms of the eigenstates |ν> of the Hamiltonian. Tip: get a recurrence relation, you will also need to use the Taylor series expansion of the exponential function. Does the eigenvalue α have to be real?
b) Find the expected value and variance of the number operator for the coherent state.
c) Calculate <α1|α_r> and Integral d²α|α><α|, where the integral is over the entire complex plane and |α> is normalized. Do these states form a complete base?
d) What is the probability of measuring n quantas (quantum) in a coherent state?
e) Does the creation operator a†(dagger) have "ket" eigenstates? And does the annihilation operator have "bra" eigenstates?