Download Exam 1 Solutions - Quantum Physics I | PHY 471 and more Exams Quantum Physics in PDF only on Docsity! Physics 471 Exam 1 Fall 2003
Name:
Mathematical formulas:
nas _ _ a n n-l
[ex sin(az) = z cos(az) + a f dz x”~ cos(az)
n = a . _h n=l ot
/ dzz” cos(az) = a sin(az) a { dz x sin(ax)
n
[dares ES =e 2 f dea" te*
a a
sin%(s) = (1- cost2e)}
cos%(z) = a+ cos(2e))
e* = cos(r) +isin(z)
1 1 1
foe — ft io f4 oft...
ea lefty hr ght ght
Infinite potential well energies and eigenfunctions:
hear?
By = pee N= 2,3->
2. (=)
4/ —sin { —
a a
Harmonic Oscillator energies and eigenfunctions:
Yn(2)
Ey = nw (n+ 5) n=0,1,2,--
(mw 4 HE) -ep fh
ba(e) = ( mh ) nl” 7 mo®
Binomial distribution:
If p is the probability of measuring £, and g = 1 — p is the probability of measuring Ep, th
probability of measuring EZ, exactly k times and Ey exactly (n — k) in n measurements is
n! kan
Plk,n—ky= in bP .
1. A particle of mass m moving in the infinite potential well
oo for «<0
V=(0 for 0<2<a
co for >a
has an initial wave function Y(z, 0) given by
A for 0O<2<a
0 for x elsewhere
U(a,0) = {
,
where A is a real constant.
(a) (1 pt) Normalize W(x, 0).
(b) (3 pt) Find (2), (x?) and o¢.
(c) (3 pt) If U(x, 0) is expanded in terms of the energy eigenfunctions as
Y(e,0) = 7 Cavala),
n=1
determine the expansion coefficients Cy.
(4) js Saitoml > jar as 2alAl®
tok
Yi40) = os xX Se
wo a
(b) 2x2 Jarai tant Z Sar
a
ss ~ gt
£45 fds x] ral? % Jace 4
te
G, += 4-4’ A
“ 3 + 2a"g
— #
ce) Asinla 2 06%
RA Whe = HE
Ae,
fale hh a ~ #&
has 2G? = 2mlod?*
Bt
ee
Rate = 57 pyr
Re
1 cot t =~ (illo?
BY
(d)
There 05 a beagnd state,
3. Give a brief but reasoned answer to each of the following:
{a) (2 pt) A particle of mass m moving in the harmonic oscillator potential V(z) =
227 /2 is in the (normalized) state
(¥o(x) + 2tdn (x) + Yo(x)]
ae
The (x) are the energy eigenfunctions of the harmonic oscillator corresponding to
the energies E, = fiw(n + 1/2). What is the probability that a measurement of the
particle’s energy will yield 3hw/2?
(b) (1 pt) The two levels, EZ, and E,, of a quantum mechanical system occur with prob-
abilities p and g = 1—p, respectively. If the energies of 5 such systems are measured,
what is the probability that EZ, will be obtained at least once?
(c) (2 pt) In a certain region of z the wave function is y(z) = Ae. What is the
probability current j(x) in this region?
(d) (1 pt) The generating function for the Hermite polynomials H,,(€) is
U(x, 0) =
efit nes
Obtain the expression for H;(€) by expanding the exponential and comparing the
coefficients of powers of ¢.
(4) P(B*#hw) = 10)? = |Ze}*=4
é
(b) Plat feast 1 &,) = J- Pl. &) = /-9
Ce) J = Mae. (aby
29)
JY = 48 As?
my
(d) ~44%2E¢
Cc = Jf + (2F£-#2 _
= D 7 4 (2€¢ 2%,
= MAE) + Het +4, a
TA
4e oF) fi TO 04 whe a COCY pC 1 Pe F of Z rs
28t, 80 Ayigy a2
