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**Unformatted text preview: **1
Self Evaluation #1 (Chem113A W'or. Fri. 020207 5:00-7:00pm+20mins. 100+5pts, max=100pts) Name: Dawl Lg By writing down my name. I conﬁrm that I strictly obey the academic ethic code when taking this exam.
ION-Ii;
(i) Five questions + one extra credit questionLPlease budget your time. You may want to start with the parts you
are more familiar with. Formula sheets are attached at the end. Need more formula? Please raise your hand.
(ii) it is very important to show calculation or algebraic details.
(iii) Theme and coverage of SE#‘l is shown below.
(iv) Academic ethics need to be strictly obeyed. No exceptions “mm” “mm” and no kidding. I Pll‘étiéi‘ ' I H all“) = Eulrl Theme of 85131. This ﬂowchart may summarize how quantum mechanics works in
modern science. The theme of this exam is to put this flowchart into practice. Step
#1, “Spectroscopic Phenomena“) “Physical Model", asks fora high level of
experience, insight, creativity, and original thinking (that's exactly why we integrate
“research fro tiers” and “advanced topics”into this coarse) and may be better left
for problen'gfat exercises. in the following, we will practice the other 3 steps. Eiocnstatcs [1] Big Picture: with GM, Who Needs QM? (total = 24 pts) Eigenvalues
[1l(a)Wann-ug (1 pts)
Relax and you will do better. In this course, we have adopted different mechanisms to make the lecturestmaterials
more entertaining and impressive, in order to communicate serious, deep concepts in QM. For example, please
circle the most relevant answer to lowingxquestion (no explanation needed): The thought experiment which
Schrodinger used to illustrate t - Inconsistency between microscopic theories and macrosclc theories is called: i I K 'SEHR-EIDINEEBS _ f an. __ . . \I L I I ‘ . - -- it, (Remark: for [1](a) and [1](b , rig m - -. . a t 1 point and thank yo answer, deduct 50 points—what were you doing in lectures?) Hub) Goals ofthis course (1 pts) - In this course. we have established ambitious goals. high standards. and solid (exhausting?) action plan in order
not only to understand the amazing microscopic world but also to apply the true understanding one day in our
careers (and if possible and lucky enough. to save a lot of lives?!) The lecture materials come from different
textbooks. research articles. and the instructor‘s exposure and research experience-in OM;"P1ease'ctrcle gne of
the following books that is least likely to be referred by the instructor least relevant to thislclourse? . , Hnoorrierii's KITTENS
u for being so engaged in lectures; wrong ” laminar" inv-
llurlnlunllledlanit __ _ ' Magic
__‘_llirncnsrons J “km Lxlq 1 c Success of CM: Bohr’s Corres ondence Princi le 4 ts _
Bohr’s “Correspondence Principle" says, “In the classical limit, quantum results approach classical results”. it
implies: in the classical limit, we do not need QM! Please use the energy density distribution p(v,T) for a blackbody
to demonstrate Bohr’s correspondence principle. That is, please prove by algebra that as T 9 00 (classical limit), 8nhv3 dv SnkT 2
_______._._.
CB e’W/kT __ 1 3 ‘v
p(v,T) = (Planck’ 5 distribution, from QM) approaches 6 (Rayleigh-Jeans
law, from CM). Please show algebraic details.
l‘v
Tit 51
Tm‘twu LXFhﬂw‘ (- ‘ ‘ + k
m, («we 3'" WW; . T
m 3 “V
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stiva 31.... )A (5111:. v
iv a 3
(.3 (v 751‘ l C
1 d Success of CM: Bohr’s model on H-like atoms 5 ts
Zc2 v2
. . . Facul: 2=mva=m€_ . . .
Usmg a combination of CM ( 47’60” r , no need to prove this) and QM (quantization of angular momentum I= mvr = nh, no need to derive this) theories, Bohr successfully explained the discrete spectrum of H-
like atoms (Nobel physics, 1922). Please derive the de Broglie matter wave relation k=hlp (where linear
momentum p=mv) based on the quantization of angular momentum I= mvr= nfi and the assumption that the quantum wave of the H-like-atom’s electron forms a circular standing wave. Please show algebraic details. .,.AM°\» smiling wa'c ‘JV 21th ’Zﬂr 17‘)‘ V) '1 : T / K"? since, +‘Ac wast”?! 5km" WV“ LC bum; h.) l ="“’" m1; Tn$c"‘l‘\':j n In“ 0.) lm’i mvr=.BUl- ' u}
)\ z\
_ b.
mwl‘ )‘
-1.
)V mv [1](e1 Downfall of CM: photoelectric effect (5 pts]. But CM totally fails to explain some important phenomena associated with microscopic objects which are the
focus of modem chemistry. For example, CM fails to explain why waves (e.g.. electromagnetic waves or light)
exhibit particle property. as shown in the “photoelectric effect" experiment. Pill The work function of platinum is 5.65 eV. What is the minimum frequency of light required to observe the
pltouoelecttic effect on Pt? 1f light with a ISO-om wavelength
is absorbed by the surface. what is the velocity of the emitted electrons? J -« .._, _- - lids 4Pts
Please se the equal partition theorem i classical me a u. o erive the molar heat capacity of a diatomic solid
such as - _.--w----. - - - - a - - - reedom for a diatomic molecule in a 3D lattice: 6 in vibrational motion
and ' rotational motion. ll l .
E-Gsc. =ik1.,c .2- 1 Downfall of CM: Planck’s formula for hoton ener 4 ts
Through studying the mystery of “blackbody radiation”, Planck proposed in 1903 that the energy of a photon can be expressed as Ephom=hv. Please solve the following question.
= Microwave radiation has a wavelength on the order of
1.0 cm. Calculate the frequency and the energy of a sin- gle photon of this radiation. Epinwa W X;
‘_ XL-
"r~~ A 1
W 339A L03 “'15 "if...
---l-~~«—u, ' ~‘(l o cm)
‘0 w” W __
3 3-0 K m Hz “3.65” 5
t z l6 3 [2] Step #2: “Physical Model” -) “Hamiltonian” (total = 16 pts)
[2](a) Schrodinger eguations (5 pts) Given the Hamiltonian H, one can write down and solve the time-independent Schrodinger equation Hu(x)=Eu(x),
where u(x) is the time-independent part of the total wavefunction LlJ(x,t). But where does this equation come from?
Please show by algebra that the time-independent Schrodinger equation can b obtained from the time- , QlIIIm t: dependent Schrodinger equation ' . Please show algebraic details and the
assumption(s) you use. q» (m5 = um t 0) Hermfﬁwi{monﬂj-é>T“WH°WI=UW-M%iuﬂ3
3* (ﬁnely :‘UDC, i HUM: .L ihﬁlﬂlﬂ: E
Um ‘Tm ’ '.Qmo=auﬁ
CW! k, 9mm fig» t (M “ |2|(b) Hamiltonian operator (5 pts) \/ QM is based upon the success of CM. For example, the Hamiltonian operator of a quantum system originates from the Hamilton function of a classical system, which is defined by Sir Hamilton as H(x,p)=p2/(2m) + V(x), where V(X) is the potential energy and p is the linear momentum. Please follow the following 3 steps to derive the Hamiltonian operator in region II for a particle in a box shown below (note: the box edge is at x=0 and x=-L) i) Write down the Hamilton function in CM for region ll; ii) Promote the CM variables into QM operators; ML iii) Use “Schrodinger representation" x = x and px = -ih d/dx (originates from “Dirac’s quantum conditions") to define the action of the Hamiltonian operator H to any function: 7
l l - ff (. I I: 111. 2‘ a;
' _ )‘
Fm V:° 1V=°Q pl) H ' m . i L
“‘L ya ‘21 " w
) H : .__.. 4 HR) Ln
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Wu, Q31) V(X)-6 _ gr (Ii/K
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H. 011 //
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A $.42. \ Q 6 [2](c) Representation (6 pts ]. Dirac’s quantum condition says that the representation of x and p can be arbitrary,
supposing that the representation satisﬁes the commutator [x, p] = in. Recall that the commutator is defined as [A,B]=AB-BA. Someone suggests the following representation:
x 9 a/ap, p -> ihp. Please prove by algebra whether this representation satisfies [x, p] = ih and is therefore a valid choice. Please show algebraic details. (Hint: Act [x, p] on an arbitrary function of linear monlentum, f(p))
Note: I d" "9 -. - -~Hl’) 7 sit-l}
[KAPILQOFX’P’HD\ 9" “9’ a» " <59 47(0)} - (Tm 4“?) Mum Nptlm‘mpprp)
: i’ﬁt‘p) ‘(ag ‘ [XIPJDHS / / [3] Step #3: “Hamiltonian” 9 “EigenstatesIEigenvalues” (total = 16 pts) [3](a) Particle in a box, region ll (10 pts). Please solve the eigenfunctions (no need to normalize) and
eigenvalues of the Hamiltonian you wrote down in [2](b) for region ll, subject to the boundary (Born) conditions for a particle in a 1D box with mass m. Please show algebraic details. Note again: the box is
located between X: -L to x=O. -« 4 I frw) 1‘3) H t :E‘ 9-: ‘ fz’x“ /' 1v». dyL .I ,, x",
3 i ,
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ﬁLde—l -~ 1t: 2 -(lsm( 1Y5 (ﬂ ; 0' § 2 32. E
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we: 6, c t‘z‘i 61 Evin»): ex”! 5163369" ISlnG uiﬂ‘ﬁ COSkX‘V C15‘”kX * /" /” unstrl' B'C C ~ 0 (-15 ’ .I \_ [3](b) Wavefunction normalization (6 pts). Please normalize the eigenfunctions you derived in [3](a) for region ll. Please show algebraic details. (Hint: You may find the integral table listed in the formula sheet helpful) 2..
{0 (-Qsmw‘gﬂk‘
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- Winter '07
- Lin
- Quantum Chemistry