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Unformatted text preview: l The formula below can be derived from equations we have discussed. 1 1 I “3,4,5, To which quantum mechanical problem does it apply?
a) blackbody radiation
b) photoelectric effect atomic hydrogen spectrum
d) particle in a box e) harmonic oscillator 2. If (/11 and gyz are normalized cigenﬁlnctions of an operator [5 belonging to different
eigenvalues ﬂ and f2 of the operator, then the integral IVI, 1/12 dx: equals all > space
0 b) 1
c) 1/5
d) ﬂ + f2 3. Which of the following is not a requirement for a valid wavefunction ‘PQQ?
a) The waveﬁmction must be an eigen function of the Hamiltonian operator.
b) For any value of X, LFog can have only one value. c) The waveﬁmction must be continuous. @ The energy of ‘i’(x) cannot be the same as that of any other valid wavefunction. 4. Foraparticleinabox,whereo<=x<=a,mstaten=3
! = (if mm“) a) What is the probability that the particle will be located at X = a/3‘?
b) Find <x> at) LVW’A) : [£)ml(3££> a, 34»
3 & $01217
W
:0
a! as
(0) MIL Ln Macy
Q/
: fl 3ﬂx
jg “(3)304 [%>M
0” ma
QJ a.
: 7a.: 0 x4144 £1 fl ,, 0b )l/7. [ l 5. Show that W306) mfg?) is a solution to the Schrodinger equation and solve for E in terms of the other constants. (121/1 87rsz
abc2 + [ 1.2 ] W 3 D mm : (5/; m (Lg)
3/17 (9/2 z w) (2%)
“(a)”; My 191‘) W
r 7— .
L“) MAKE/X) + WW") :
wan”) : 0‘“ 0v 6. The angular momentum operators are: I; z * mp f: a J x az "’zé—y“
‘ _ _ 5 6 A . a a Showthat [i,,Ly]= m L Some Postulates and General Principles of Quantum Mechanics 93 ___________________._..———————————— 4—17. Referring to Table 4.1 for the operator expressions for angular momentum, show that [12“ ﬂy] 2 mil try, L1] 1?th and
[if 12x1=miy (Do you see a pattern here to help remember these commutation relations?) What do these
expressions say about the ability to measure the components of angular momentum simultaneously? H __. 11 gay
L*L"f’l ‘hlyaz Zaylll ‘hlzax "azll :_h2(y3f 6f x 32f 23—2f_ x 32f) ' f In the same way, we can show [11)” £1] = ihix and [1:3 11;] = [h if. The pattern involves the
cyclic permutation of x, y, and 1. Since no combination of the operators ix, 1),, and £1 commutes,
it is not possible to simultaneously measure any two of the three components of angular momentum
to arbitrary precision (as discussed in Section 4—6). ...
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This note was uploaded on 10/18/2011 for the course CHEM 110A taught by Professor Galli during the Fall '10 term at UC Davis.
 Fall '10
 galli
 Physical chemistry, pH

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