Download Quantum Theory 1 - Class Exercise 4

Quantum Theory 1 - Class Exercise 4
1. Consider a Hamiltonian which describes a one dimensional system of two particles of masses
m1 and m2 moving in a potential that depends only on the distance between them.
Ĥ =
p21
p2
+ 2 + V (x1 − x2 )
2m1 2m2
(a) Write down Schroedinger’s equation using the new variables
X=
m1 x1 + m2 x2
m1 + m2
,
x = x1 − x2
. Interpret the meaning of these variables.
(b) Use seperation of variables to find the equations of motion of X and x.
2. Consider a ”quantum dice”, which is just a quantum description of a regular playing dice.
We define the number operator N̂ ϕn = nϕn and the evenness operator Ẑϕn =
1+(−1)n
ϕn .
2
(a) When measuring the number operator, what is the probability of getting N̂ = 4?
(b) When measuring the evenness operator, what is the probability of getting Ẑ = 1?
(c) We measure the number operator and get N̂ = 4. Afterwards, we measure the evenness,
what is the value we are going to get?
(d) We measure the evenness operator and get Ẑ = 1. Afterwards, we measure the number
operator. What is the probability of measuring n = 4?
3. Consider a free particle in an infinite well of length L. Find stationary states, eigenenergies,
and eigenvalues of the momentum operator.
4. Consider a free particle in an infinite well of length L. At time t = 0 the particle is prepared
in the state
√
√
√
ψ(x, 0) = A ϕ1 + 2ϕ2 + 3ϕ3 + 2ϕ4 + ϕ5
(a) Find |A|.
(b) Find ψ(x, t).
(c) What is the probability of measuring an energy larger then
1
2~2 π 2
mL2
(d) What is the probability of measuring a momentum of
4π~
L
(e) In a measurement of the particle’s energy we got the value
we measured the particles momentum.
• What is the probability of having p =
2π~
L ?
• What is the probability of having p =
4π~
L ?
2
4~2 π 2
.
mL2
After this measurement,