# Exercises in Theoretical Physics II - Quantum Mechanics Set 2 by olliegoblue30

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Institute for Theoretical Physics                                                            SS09
prof. Milena Grifoni

Exercises in Theoretical Physics II - Quantum Mechanics
Set 2 (28.04.2009)
Assistants:
Dr Magdalena Marga´ska-Ly˙ niak (3.1.22, phone 943-2042)
n      z                                    Wed 15-17 Phy 5.1.01
Dr Silvia Garelli (3.1.25, phone 943-2039)                         Wed 13-15 Phy 5.1.01
Dr Alex Matos Abiague (3.1.32, phone 943-2028)  Wed 13-15 Phy 9.1.11, Fri 8-10 Phy 5.1.01

Problem 2.1: Gaussian wave packet of a free particle
At initial time t = 0, a normalized wave packet of a free particle with mass m in one dimension
is given as
1             x2
ψ(x, 0) =    √   exp − 2 + ik0 x .
a π           2a
o
The evolution of the wave packet is determined by the one-dimensional Schr¨dinger equation
for a free particle.
a) Calculate the Fourier transform of the initial wave packet ψ(x, 0), (2 points)
∞
1
ψ(k, 0) = √               ψ(x, 0)eikx dk.
2π       −∞

o
b) Derive the Schr¨dinger equation for the wave packet in momentum space, ψ(k, t) with the
o
help of the Schr¨dinger equation for a free particle. Solve the equation with the initial condition
on ψ(k, 0) obtained in a). (2 points)
c) Show that the time-dependent wave packet ψ(x, t) has the expression (2 points)
1              (x − v0 t)2
ψ(x, t) =    √             exp − 2             + i(k0 x − ω0 t) ,
a π(1 + it/τ )      2a (1 + it/τ )
where
2
k0           k0           ma2
ω0 =      , v0 =       , τ=          .
2m           m
d) (To be discussed in class (2 points) ) The average position of a particle at time t is deﬁned
as                                          ∞
xt=        x|ψ(x, t)|2 dx.
−∞
Show that
x t = v0 t.
e) (To be discussed in class (2 points) ) The standard deviation of the particle position is deﬁned
as
∆xt =      (x − x t )2 ,
and the variation of the particle position can be calculated as
∞
2
(x − x t )    =        (x − x t )2 |ψ(x, t)|2 dx.
−∞

1
Show that
2
a            t
∆xt = √      1+             .
2          τ
f) Sketch the wave function and its absolute value at time t = 0 and t = τ . (2 points)

Problem 2.2: Wave packet in momentum space
Consider a 1D wave packet given in the momentum space by the function

ϕ(p) = A θ( /d − |p − p0 |).

a) Calculate the normalization factor A. (2 points)
b) Calculate the function ψ(x, 0) in the position space. Hint: remember that p = k.
What is the form of ψ(x, t) in the limit of /d     p0 ? (2 points)
c) Calculate x , p , ∆x and ∆p in the limit of /d         p0 . (2 points)

Maximum total points: 18

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