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Unformatted text preview: ll ' 25. Particle in a 2D box (15 points) Name: A particle of mass, m, is conﬁned to a rectangular 2D box that measures L in the x—direction and
2L in the y—direction. The walls of the box are inﬁnitely high, thick, and inﬁnitely repulsive. The
motion of the particle in the two dimensions is independent. a) From the solution of Schrodinger’s equation for the particle in the 1D box write down an
equation for the energy levels in the two dimensional box. You must specify the values that any quantum numbers can have. ,
b) Draw an energy level diagram (arrangement of energy levels in order of increasing
energy) for the ﬁrst six energy levels of two dimensional box. Make sure to label all levels with the appropriate quantum numbers and draw the energy scale appropriately.
c) Assuming the particle is initially in the ground state, draw an absorption spectrum for the
. particle in this 2D box. Label axes and be as accurate as possible in drawing the spectrum toscale. , i I 7/ '7’ L 2 'L
3F a) E ’ ”Zl‘ +33"? 3L zlh +12?)
/ "’k S'MLZ 8mm gmt , Lt
/ " L n’, ,7L23; tut: L1 lLlWZ+k) ,’ l2; \,L,'§)Lf:,.. 3Zn L h: t
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20 176 LL; 2/) AW r" ("'3 . 5845 n 2/: E”
o . +mta Mat Mme, mt mm. 12 l .26. BONUS QUESTION (10 pts.) Name: Betacarotene is found in carrots and many other orange colored vegetables. It has the
structure shown below. * Thismolecule absorbsat ~ 450 nm. Let us assume that a particle in the box is a
reasonable model for an electron in this molecule. ' a) What size box would be consistent with an absorption at 450 nm? Think careﬁilly
about what transition (i.e. which quantum numbers) is involved. b) If the length of a C—C bond is 1.54 angstroms and the C=C is 1.34 angstroms, does
your answer make sense? (langstrom = 1 X 10'10 m) JUstify either the agreement or
disagreement of your answer with the particle in the box result. ...
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 Fall '11
 StephenJacobson

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