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PHY 3063 Problem Set
Due Tuesday April 24 (in class)
(Total Points = 105)
Reading: Read Tipler & Llewellyn Chapter 7.
Problem 1 (20 points): The Pauli spin matrices are given by
σ x
i
i
σ y
σ z
(a) (10 points) Show that σ↑i = σi , det(σi) = -1, Tr(σi ) = 0, [σi ,σj ] = 2iεijkσk, and {σi ,σj } = 2δij. Note that [A, B] = AB – BA and {A, B} = AB +BA. (b) (10 points) Show that
l
σ (^) i σ j δ ij i ε ijl σ l.
Problem 2 (20 points): Quarks and anti-quarks carry spin ½. (a) (10 points) Three quarks bind together to form a baryon (such as a proton or a neutron). What are the possible spin states for a baryon (assuming that the quarks are in the ground state so that the orbital angular momentum is zero). (b) (10 points) A quark and anti-quark bind together to form a meson (such as a π-meson). What are the possible spin states for a meson (assuming that the quarks are in the ground state so that the orbital angular momentum is zero).
Problem 3 (10 points): Evaluate the following in SU(2). (a) (1 point) 2 × 1 = (b) (1 point) 2 × 2 = (c) (1 point) 3 × 2 = (d) (1 point) 3 × 3 = (e) (1 point) 5 × 2 = (f) (1 point) 5 × 3 = (g) (1 point) 4 × 2 = (h) (1 point) 2 × 2 × 2 = (i) (1 point) 2 × 2 × 3 = (j) (1 point) 3 × 3 × 3 =
Problem 4 (55 points): Consider the case of two non-interacting particles both with mass m in a one-dimensional infinite square well given by V(x) = 0 for 0 < x < L, and V(x) = ∞. For one particle we know that (see problem set #2) the stationary states of Schrödinger’s equation are given by
( ,) ( ) iEt /^ h n n
Ψ x t =ψ xe − n with^ ( )^2 sin( n x / L )
L
ψ n x = π
and E (^) n = n^2 E 0 , and n is a positive integer and (^2)
2 2 E 0 (^) (^2) mL
π h =. For two ( non-interacting ) particles
we look for a solution of the form
0 L
Two Particles in a Box
ψ (x 1 )
ψ (x 2 )
h / h 1 2
/
Ψ x x t =ψ x x e − iEt^^ =ψ x ψ x e − iEt with E
m
p m
p (^) x x
(a) (5 points) Work out the energy levels, E αβ , and the normalized eigenfunctions, ψ αβ( x 1 , x 2 ),
for the case where the two particles are distinguishable. What is the ground state energy (in terms of E 0 )? What is the energy of the 1 st^ excited state (in terms of E 0 )? Construct the
probability density ρ αβ D ( x 1 , x 2 )(D = distinguishable) and show that
0 0 1 2 1 2
L L (^) D ρ αβ x x dxdx
(b) (5 points) Work out the energy levels, E αβ , and the normalized eigenfunctions, ψ αβ( x 1 , x 2 ),
for the case where the two particles are identical bosons. What is the ground state energy (in terms of E 0 )? What is the energy of the 1 st^ excited state (in terms of E 0 )? Construct the
probability density ρ αβ BE ( x 1 , x 2 )(BE = Bose-Einstein) and compare it with the “classical”
probability density for two identical particles,
ραβ classical ( x 1 , x 2 )= 21 (ραβ D ( x 1 , x 2 )+ρ αβ D ( x 2 , x 1 )).
What is
L L (^) BE
x x dxdx
0 0 1 2 1 2 ρ αβ ( , )?
(c) (5 points) Work out the energy levels, E αβ , and the normalized eigenfunctions, ψ αβ( x 1 , x 2 ),
for the case where the two particles are identical fermions. What is the ground state energy (in terms of E 0 )? What is the energy of the 1 st^ excited state (in terms of E 0 )? Construct the
probability density ρ αβ FD ( x 1 , x 2 )(FD = Fermi-Dirac) and compare it with the “classical”
probability density for two identical particles, ρ αβ classical ( x 1 , x 2 ). What is
L L (^) FD
x x dxdx
0 0 1 2 1 2 ρ αβ ( , )?
(d) (5 points) For the ground state and the 1st^ excited state, what is the probability of finding both particles in the left half of the square well (0 < x < L/2) if the two particles are
distinguishable ( i.e. using ρ αβ D ( x 1 , x 2 ))?
(e) (5 points) For the ground state and the 1st^ excited state, what is the probability of finding both particles in the left half of the square well (0 < x < L/2) if the two particles are “classically”
identical ( i.e. using ρ αβ classical ( x 1 , x 2 ))?
(f) (5 points) For the ground state and the 1st^ excited state, what is the probability of finding both particles in the left half of the square well (0 < x < L/2) if the two particles are identical
bosons ( i.e. using ρ αβ BE ( x 1 , x 2 ))?
(g) (5 points) For the ground state and the 1st^ excited state, what is the probability of finding both particles in the left half of the square well (0 < x < L/2) if the two particles are identical
fermions ( i.e. using ρ αβ FD ( x 1 , x 2 ))?
(h) (5 points) For the ground state and the 1st^ excited state, what is the average separation between the two particles squared, <(x 2 -x 1 ) 2 >, if the two particles are distinguishable ( i.e. using
ρ αβ D ( x 1 , x 2 ))?
