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MASSACHUSETTS INSTITUTE OF TECHNOLOGY
5.73 Quantum Mechanics I Fall, 2002
Professor Robert W. Field
Problem Set #
DUE : At the start of Lecture on Friday, September 20.
Reading : Merzbacher Handout, pp. 92-112.
Problems :
a^2
- (^) ψ 1 ( )x = ,
b x − x 0 )^2 + c^2
2 a b, and c are real
∞ A. Normalize ψ 1 ( ) in the sense x (^) ∫−∞ | ψ |^2 dx = 1.
B. Compute values for x , x^2 , and ∆ x for ψ 1 ( x ).
C. ( optional ) Compute values for k and ∆ k for ψ 1 ( k ), where ψ 1 ( ) is the k Fourier transform of ψ 1 ( x ). [If you choose not to do this problem, state what you expect for the form of Ψ 1 (k) and the magnitude of ∆k.] (^2) (
- ψ 2 ( ) x = e − c^ x^ − b )
2 ei α( x )^ where c , b , and α( x ) are real. Use the stationary phase idea to design α( x ) in the region of x near x = b so that k = k 0 ≠ 0.
- Merzbacher, page 111, #2.
- The following problem is one of my “patented” magical mystery tours. It is a very long problem which absolutely demands the use of a computer for parts F and G. There are many separate computer programs that you will need to write for this problem. I urge you to divide the labor into smaller groups, each responsible for a different piece of programming. I believe that the insights you will obtain from working together on this problem will be more than worth the effort expended.
Consider the simplest possible symmetric double minimum potential:
V( x ) = aδ( x ) a > 0 –L/2 < x < L/ V( x ) = ∞ | x | ≥ L/2.
Problem Set #
A. Solve for all of the eigenstates and eigen-energies for states that have odd reflection symmetry about x = 0. (This part of the problem is very easy.)
B. Solve for the energy eigenstates and eigen-energies for the 5 lowest energy even-symmetry states. Choose a = 400h^2 /Lm. I suggest you use trial functions of form
ψn( x ) = N sin[kn( x + L/2)] –L/2 ≤ x < 0 ψn( x ) = –N sin[k n( x – L/2)] 0 < x ≤ L/
One way to find the eigen-energies is to plot the quantities y = tan(kL/2) and y = –kL/400 and to determine eigen-energies from the k-values at intersections. Each En (odd n, even symmetry) is located at an intersection. Note there will be exactly one value of En below the lowest odd-symmetry eigenstate (E 2 ) and one value of En between each consecutive pair of odd- symmetry eigenstates.
C. For an ordinary infinite square well. the ratio of the spacing between the two lowest levels to that between the two lowest odd–symmetry levels, is
R^ E^2 −^ E^1 4 −^1 = 3.. 21;42 ≡^ E 4 −^ E 2
= 16 − 4 12
= 0 25
For your double minimum potential, this level spacing ratio will decrease from 0.25 at a = 0 toward 0 as a increases. For the value of a that I suggested, this ratio should be about 0.003.
Repeat the calculation of R21;42 for E 1 using a -values a factor of 3 and 9 smaller than the one you decided on above.
Suggest a functional relationship between a and R21;42.
D. The ratio
R43;42 = 7/
for an ordinary infinite square well. Is the E 4 -E 3 2 –E 1? Why?
spacing you obtained for a = 400h^2 /Lm larger or smaller than E
Problem Set #
(^2 ) (i) Plot Ψ L ( , 0) x 2 , Ψ L (^)
x , 8 mL
(^2) ^ ^ , and^ Ψ L^ ^
(^) x ,
( E 2
h
− E 1 )^
h Comment on what you see in these 3 plots. There is a huge amount of information. “Assign” as many features or families of features as you can.
(ii) Calculate the following quantities and plot the following quantities twice, once over a short 0 ≤ t ≤ 2tg and once over a long 0 ≤ t ≤ tt time interval,
x t = ∫ Ψ L ( , ) L ( x , t ) dx
- (^) x t x Ψ x (^) L ( x , t ) dx (^2) t =
∫ Ψ L^ ( , )^
2
2 1 2^ /
∆ x t = [ x^2 t − x t ].
(iii) Compare 〈x〉t and ∆xt crashes toward 0. What might account for such a focussing of the wavepacket?
and explain why the position variance exhibits periodic
