3) Considerabunch of electrons. Each has this wave function, a superpositionoftwo different Y_l,m:

Ï =aY_l1,m1 +bY_{l2, m2}
Suppose you know (by makingmany(l,m) measurements)that the probability of obtaining (l 1, m1)is 1/4 and the probabilityof obtaining (l 2, m2) is 3/4. Which ofthese setsof (a,b) values will give you theobservedprobabilities?
(a,b) = (1, 3)
(a,b)= (1/4, 3/4)
(a,b)= (1/2, â3/2)

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Part b: Collision in Two-Dimensions

The law of conservation of momentum is true for any and everydirection. So, for an isolated system, the momentum in the x, y andz directions or any other direction does not change.

Consider two objects colliding in deep space. Since the two objectstogether can be considered an isolated system, the total momentumof the system (the sum of the momenta of the two objects) isconserved in both the x and y directions. That is

pi-x-total=pf-x-total or pix1+pix2=pfx1+pfx2

pi-y-total=pf-y-total or piy1+piy2=pfy1+pfy2

If we assume object one is moving in the x-direction, object two ismoving in the y-direction and the collision is perfectlyinelastic,
we can rewrite these equations as

m1*vix1=(m1+m2)vfx

m2*viy2=(m1+m2)vfy

These equations can be solved to find the x and y components of theobject's velocity after the collision. The magnitude of vf can befound using the Pythagorean Theorem. The direction of the objectsafter the collision can be found using the inverse cosinefunction

Î¸=cos-1(vfx/|vf|)

Since cos-1 outputs an angle between 0 and Ï radians, we must useÎ¸new=2Ï-Î¸old if vfy is negative to get a correct value. Use thebuilt in function 'if' to find the true value of Î¸. Afterwards,convert the angle into degrees using the built in function'degrees' .

i) Fill in the missing values in the table on the bottom usingformulas to find each value

This is the half of the tablethat is full so far

A = Mass1

B = Velocity initial 1

C = Mass2

D = Velocity intial 2

E = M2/M1

F = Vi2/Vi1

This is therest of the chart that needs formulas

G =Pi-x

H =Pi-y

I =|Pf|

J =Vf-x

K =Vf-y

L =|Vf|

M =Î¸

N = Î¸ if Vi2< 0

O = Î¸(degrees)

A Particle in a box
In order to answer the following questions you will need toconsideran infinite potential
well of length L, in which a particle has been confined.

1. For principal quantum number n=7, write code to calculatethenormalization constant.
Then, plot the probability density function for this state (usingL= 10).
2. Using the properly normalized wave equation from Question1,write code to determine
the expectation values <x> and <x2> . Is <x>oneof the most probable positions? Explain
why.
3. Using the properly normalized wave equation from Question1,write code to determine
the expectation values <p> and <p2> . Show the valueyouobtain for <p2> is equal to
2mE7 (where E7 is the energy of the n = 7 level). Remember that pisa differential
operator so you will have to use the diff command.
4. The uncertainty of a quantum measurement is defined on page222,Equation 6.41 (3rd
edition SMM). Using this equation calculate the uncertainties oftheposition and
momentum of a particle in the n = 5 state. Compute the productofthese uncertainties and
compare this with the prediction of the Heisenberguncertaintyprinciple.
Quantum oscillator
5. Repeat calculations in the previous question for the groundstatewave function of the
quantum oscillator. Comment on the obtained result.

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