Physics 741 Graduate Quantum Mechanics 1
Solution Set Q
1. [10] In chapter nine, section A, we were searching for matrices D ( R ) which satisfy
D ( R1 ) D ( R2 ) = D ( R1R2 )
One easy way to make this equation work out is to define
D (R) = R
Our goal in
Physics 742 Graduate Quantum Mechanics 2
Solution Set Q
1. [15] A spin particle of mass m lies in a one-dimensional spin-dependant potential H0 = P2 1 + m 2Q 2 P+ 2m 2 where 1 0 P+ = . 0 0
In other words, the potential only affects particles in a spin-up
Physics 741 Graduate Quantum Mechanics 1
Solution Set P
1. [30] In class we solved the hydrogen atom, which is a spherical 1/r potential. Consider another spherically symmetric potential, namely, the spherical harmonic oscillator H= P2 1 + m 2Q 2 2m 2
Thi
Physics 742 Graduate Quantum Mechanics 2
Solution Set P
1. [25] A hydrogen atom is in interstellar space in the 1s state, but not in the true ground state (F = 0), but rather in the hyperfine excited state (F=1), specifically in the state I = n, l , j , F
Physics 741 Graduate Quantum Mechanics 1
Homework Set O Due Wednesday, October 15
1. [10] Two particles interacting via a spherically symmetric relative potential have Hamiltonian
P12 P22 H= + + V ( Q1 Q2 2m1 2m2
Define new operators
)
P P1 + P2 Q ( m1Q1
Physics 742 Graduate Quantum Mechanics 2
Solution Set O
1. [20] A hydrogen atom is in the 1s ground state while being bathed in light of sufficient frequency to excite it to the n = 2 states. The light is traveling in the +z ij direction and has circular
Physics 741 Graduate Quantum Mechanics 1
Solution Set N
1. [10] Let J = J x , J y , J z be three Hermitian operators that commute with the Hamiltonian H, and have angular momentum-like commutation relations
(
)
J x, J y = i Jz,
Define the operators
J y ,
Physics 742 Graduate Quantum Mechanics 2
Solution Set N
1. [15] A particle of mass m is initially in the ground state of a harmonic oscillator with frequency . At time t = 0, a perturbation is suddenly turned on of the form W ( t ) = AQe t . At late times
Physics 741 Graduate Quantum Mechanics 1
Solution Set M
1. [20] A particle of mass m and energy E in two dimensions is incident on a plane step function given by
0 if Qx < 0, V ( Qx , Qy ) = V0 if Qx > 0.
The incoming wave has wave function in ( x, y ) =
Physics 742 Graduate Quantum Mechanics 2
Homework Set M
1. [5] Tritium is 3H an isotope of hydrogen which radioactively decays to 3He. Assume an isolated 3H has its electron in the ground state when it suddenly radioactively decays to 3He, but the nucleus
Physics 741 Graduate Quantum Mechanics 1
Solution Set L
1. [10] In class we found the even parity bound states for H= P2 + V (Q ) , 2m where
0 V (Q ) = V0
if Q < a if Q > a
Repeat the analysis for the odd parity bound states.
We know from class notes eq
Physics 742 Graduate Quantum Mechanics 2
Solution Set L
1. [25] A particle of mass m scatters from a potential
V ( r ) = F ( r a )
so that the potential exists only at the surface of a narrow sphere of radius a (a) [4] What equation must the radial wave f
Physics 741 Graduate Quantum Mechanics 1
Solution Set K
1. [10] At t = 0, a single particle is placed in a harmonic oscillator
H = P 2 2m + 1 m 2Q 2 2
in the superposition state
3 4 (t = 0) = 5 1 + 5 2 ,
That is, in a superposition of the first and second
Physics 742 Graduate Quantum Mechanics 2
Solution Set K
1. [15] An electron of mass m scatters from a neutral hydrogen atom in the ground state located at the origin. (a) [7] What is the charge distribution for a neutral hydrogen atom? Dont forget the nuc
Physics 741 Graduate Quantum Mechanics 1
Solution Set J Due Friday, September 26
1. [15] A particle of mass m lies in a one-dimensional infinite square well in the region [0,a]. At t = 0, the wave function in the allowed region is given by x ( t = 0 ) = 3