Phy4604 Quantum Mechanics I
Math practice I
Peter Hirschfeld
Fall 2004
1. Linearity. Given the following operators, which are linear?
(a) ψ → x3 ψ
d
(b) ψ → x dx
ψ
(c) ψ → λψ ∗
(d) ψ → eψ
d
dx ψ + const.
Rx 0
(f) ψ → 0 dx ψ(x0 )
(e) ψ →
2. Commutators. Calculate the commutators (L± are the SHO ladder
operators defined in class):
d
(a) [x3 , x dx
]
(b) [∇2 , 1r ]
d
]
(c) [x2 , dy
(d) [3, x2 ]
d
(e) [3, dx
]
(f) [L+ , L− ]
(g) [L+ , L+ ]
(h) [L+ , x]
(i) [L+ , p]
(j) Given three operators A, B, and C, express [AB, C] in terms of [A, C]
and [B, C].
3. Complex numbers. Simplify:
(a) |3 + 2i|2
p
(b) Re x2 − 2ixy
(c) Im eiωt/h̄
(d) Re
x2 −2iz
z 3 −2ix
1
4. Fourier transform. For these problems assume the asymmetric normalization of the Fourier transform:
Z
f (x) =
g(k)eikx
g(k) =
=
1
f (x)e−ikx
2π
Try to do each of these analytically, then check with Maple. Hint: some
may be easier if you use contour integration.
(a) f (x) = “window” function of width a,
½
f0 |x| < a/2
f (x) =
0 |x| > a/2
g(k) =?
(b) f (x) = Gaussian exp(−x2 /(2σ 2 ), g(k)= ?
(c) f (x) = A exp(−a|x|), g(k) =?
(d) Lorentzian f (x) =
A
(x2 +a2 ) ,
g(k) =?
5. Projecting out amplitudes. Determine the amplitude desired in each
problem:
(a) Expand window function above in 1D SHO eigenfunctions. What is
the amplitude of the 1st excited state?
(b) Expand the sawtooth function
f (x)

 x + a/2 < −a/2 < x < 0
x
0 < x < a/2
=

0
otherwise
in terms of eigenstates of the infinite square well between (−a/2, a/2).
Determine the amplitude of the 1st and 2nd excited states.
6. Delta functions. Evaluate:
R3
1 2
(a) −3 dx δ(x − 2) exp− 2 x
R3
1 2
(b) −3 dx δ(x − 4) exp− 2 x
R
(c) d3 r eik·r e−ik·r
R
(d) d3 r eiak·r e−ibk·r
7. Inner products.
2
(a) Simplify (αA + βB, γC + δD), where Greek letters are complex constants and Roman letters are operators.
(b) Evaluate (ψ0 , x2 ψ2 ) for SHO wavefunctions.
(c) Evaluate (ψ1 , x2 ψ2 ) for SHO wavefunctions.
8. Hermitian operators. Let x and p be the usual operators for position
and momentum. State whether each of the following operators is selfadjoint (Hermitian), anti-self-adjoint (antiHermitian, O† = −O), unitary
(O† O = 1), or if none of the above, what the adjoint is. Hint: remember
(AB)† = B † A† .
(a) xxp
(b) xpx
(c) xpp + ppx
(d)
d2
dx2
(e)
d3
dx3
p
(f) e
9. More operators. Two operators A and B satisfy A = B † B + 3 and
A = BB † + 1.
(a) show A is self-adjoint
(b) find [B † , B]
(c) find [A, B]
(d) Suppose ψ is an eigenfunction of A with eigenvalue α. Show that if
Bψ 6= 0, then Bψ is an eigenfunction of A, and find the eigenvalue.
10. Gram-Schmidt. The Legendre polynomials P` (x) are a set of real polynomials orthogonal to each other on the interval −1 < x < 1:
Z
1
dxP` (x)P`0 (x) =
δ`,`0 .
−1
The polynomial P` is of order `, i.e. the highest power of x is x` . Starting
with the set of functions ψ` (x) = x` , use the Gram-Schmidt procedure to
construct P0 (x), P1 (x), and P2 (x).
3