Download Assignment 3

Chennai Mathematical Institute
QUANTUM MECHANICS
Problem sheet 3
Due date: 30th August 2007
13. Generalise the Born’s probability to a two particle system. Then generalise it for a N particle system ( in 3 dimensions).
14. Generalise the time dependent Schroedinger quation to a two particle
system. Solve it for two particles of masses m1 and m2 confined in a box of
length L. In particular obtain allowed values of energy and the coresponing wave
functions.
15. Calculate the average value of x, x2 , p, p2 for a particle in a box having
2 2 2
π n
energy En = h̄2mL
2 . Calculate ∆x∆p for the state where
∆O = (< O2 > − < O >2 )1/2
where O stands for x or p.
16. If the wave function of the particle is given by
ψ(x, t) = A(2/L)1/2 sin(πx/L)e−iE1 t/h̄ + B(2/L)1/2 sin(2πx/L)e−iE2 t/h̄
2
2 2
π k
where Ek = h̄2mL
2
Find the average position of the particle.
17 (a)Calculate the comutators:
(1)
xpn − pn x
(2)
xn p − pxn
(b) For a three dimensional we have
[xi , xj ] = [pi , pj ] = 0
and
[xi , pj ] = ih̄δij
where xi , pi are the i-component of position and momentum rspectively.
~ = ~r × ~p
(c) Angular momentum is defined by L
Show that
[Lj , Lk ] = ih̄ǫjkm Lm
18. A particle is confined in a three dimensional box of sides A,B and C.
Find the energy levels an the corresponding time independent wave functions.
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