1
The following wave function (in which A and B are numbers not functions of x or
E1 t
E1 t
t) (x, t) = A1 (x)e ~ + B2 (x)e ~ is a linear combination of two solutions of the time
En t
dependent Schrdinger equation each having the form n (x)e ~ with n=1 an
1
Using the ve-term linear trial function
in Table I below
i
1
2
3
4
5
=
q5
i r2
i=1 Ci e
with the exponents
i
33.865
5.09479
1.15879
0.32584
0.1027410
Solution:
Mathematica Code:
Clear[i, j, n, A, H, T, V, S, eig, val];
A = cfw_33.865, 5.09479, 1.15879,
1-7
Sirius, one of the hottest known stars, has approximately a blackbody spectrum
with max = 260nm. Estimate the surface temperature of Sirius.
Solution:
T
=
= T =
hc
4.96511kT
2.8977685103 m K
260109 m
hc
4.96511k
(6.626069571034
J s)(2.99792458108 m/s
1a
Using atomic units, for the quantities in the general formula for the energy
Z2
e2
levels of a one electron atom, En = 2n2 a0 40 , show that the energy of the helium ion
(He+ , with Z = 2) in its ground state (1s state) in atomic units is -2. These ene
1a
Show that
Solution:
is 0.
s
x2 dx
xe
=0
The integrand is an odd function, and it is integrated over symmetric limits, so the answer
Beyond that, to evaluate over arbritary limits, let u = x2 = du = 2xdx = xdx =
sb
s x=b
s b2
s
2
2
= eu du = a xeax dx
1
Find the values of the following commutators for the hydrogen atom Hamiltonian. Only very little algebra is required, because you may use the values of any of the
angular momentum commutators we wrote in class without deriving them.
(a) H, Lx (b
1
The internuclear distance of the 12 C 16 O molecule is 1.1283 angstroms. Look
up the masses of these isotopes of the atoms on the NIST website(http:/physics.nist.
gov/PhysRefData/Compositions/index.html) and use this data to compute the rotational
const