Obtain an expression for the expectation value in terms of energy

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Homework Statement

The ground state wave function for a particle of mass m moving with energy E in a one-dimensional harmonic oscillator potential with classical frequency omega is:

u(subscript 0) (x)= N(subscript 0) exp((-alpha^2)(x^2)/2) and alpha=sqrt (m *omega/h-bar)

where N(subscript 0) is some normalisation constant

a) for the ground state, show explicitly that the quantum mechanical expectation values <x> and <p> are both zero
b) if the uncertainties delta x and delta p are given by:

(delta x)^2=<x^2>-<x>^2
(delta p)^2=<p^2>-<p>^2

obtain an expression for the expectation value of E in terms of the uncertainties
c)If ΔxΔp = c for c a constant, deduce a value for c by minimising the ground state energy. What significance
does the value of c have in light of the uncertainty principle ?

The Attempt at a Solution

I haven't tried part a yet, but for part b:

<E>=<p^2>/2m
=((delta p)^2 - <p>^2)^2)/2m

I don't know if the above is correct and I have no idea how to get it in terms of delta x. Please help

 

[dextercioby]

The expectation value (average over an ensemble) for the Hamiltonian is

$$\langle H\rangle =\frac{1}{2m} \langle p^2\rangle + C \langle x^2\rangle$$

, wherer C=C(m,ω)>0.