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Tutorial 02 April
(a)
T T T
(b)
2
2
2 2
[ ˆ^ , ˆ^ ] ˆ ˆ^ ˆ ˆ
x y x y y x
x y
y x
x y y x z
s s s s s s
i i s s i i
i s s i
i (^) i s s s s i s i
^
(c) Similarly
(d)
z
z
s
s
^
, etc
(e) Similarly
Each function has two parts, a spatial and a spin part. Determine the symmetry for each part. IF (+)
means symmetyc, and (-) means anti-symmetric, then (+)(+) = (+), (+) (–) = (–) and (–)(–) = (+). This
results in a (–), b(+), c(+), d(–) and e (neither)
Extra question
A researcher explores the trial function
2 ( )
x x e
ψ to calculate the energy of the one-dimensional
harmonic oscillator subject to the condition that V(x) = 0, i.e. the Hamiltonian is
2 2
2
d H m dx
(i) Derive an expression for the expectation value (average value) of the energy, E.
(ii) Assuming that = n ( n -1), for which value of n will the energy, E , have its lowest value?
(i)
2
2 2
1/
1/ 2
2
E
m
x
E
m
ψ(x)|H|ψ(x) ψ(x)|ψ"(x)
ψ(x)|ψ(x) ψ(x)|ψ(x)
ψ"(x) ψ(x)
ψ(x)|ψ"(x)
ψ(x)|ψ(x)
(ii) Set
2
n n , then calculate 0
E
n
, and solve for n. This gives n = ½.
