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.اولا : Average momentum :

(a) A particle’s coordinate space wavefunction is square-integrable and real up to an arbitrary multiplicative phase:
ψ(x) = exp( iα)φ(x
b) Now suppose that α→α(x) (ie. α varies with position), but is still real. What
is the average value of the momentum?
ثانيا :
(Properties of a wavefunction )
A) particle of mass m moves on the line x ∈ [−∞, ∞], and has the following wave function at some time:

ψ(x)= N cosbx
0 for |x|< π/2b


0 for |x| >π/2b صفر =
(a)
Normalize ψ(x). That is, find the value of N .
(b)
What is <x>? What is <x2>? Is ψ(x) a position eigenstate?
(c)
What is <p>? What is <p2>? Is ψ(x) a momentum eigenstate?
(d)
If the momentum of ψ is measured, what is the probability distribution, P(p), of the results? Sketch P(p).
(e)
Suppose ψ(x) describes a free particle, so V (x)=0. Is ψ(x) an energy eigen-state?
3. Properties of a another wavefunction (14 points)
A particle of mass m moving under the influence of a one-dimensional potential V (x) has the wave function:
ψ(x) = N x exp(−αx 2) (2)

Normalize ψ(x). Assume that α > 0 (a)

(b)
Is ψ(x) a position eigenstate? Is ψ(x) a momentum eigenstate? Explain your reasoning.