Scattering in QM

Consider a classical point particle of mass $m$ incident on a large hard sphere of radius R as shown. Suppose that the colission with the sphere is elastic and the mass of the sphere is much greater than that of the incoming particle. The impact parameter $b$, the initial velocity$v_i$ and the positive $z$ direction are also indicated.

a) Determine the magnitude and direction of the initial angular momentum of this particle in terms of given quantities. What is the $z$ component of this angular momentum?

b) Is the magnitude of the angular momentum of the outgoing particle greater than, equal or les than the magnitude of the angular momentum of the incoming particle. Explain?

c) Determine the direction of the outgoing angular momentum of this particle. What is the $z$ component of this angular momentum?

Suppose the impact parameter $b$ were decreased.

d) Would this change the magnitude of the angular moment of this particle?

e) Would this change the direction of the angular moment of the particle?

Consider a quantum mechanical particle with the wave function $psi_{in}(r,theta,phi)=Ne^{ikz}$ that is incident on a spherically symmetrical potential.

f) Which of the following quantities is well defined for the incident particle?

i) The magnitude of linear momentum $p$. Explain.

ii) The orbital angular momentum $L_z$. Explain.

iii) the total angular momentum squared $L^2$. Explain.  
iv) Write the incident wavefunction $psi_{in}(r,theta,phi)=Ne^{ikz}$ for this particle in terms of a superposition of $|klm>$ where $l$ and $m$ are quantum numbers associated with $L^2$ and $L_z$ respectively. Do the coefficients in your expression depend on $k$ , $l$ or $m$? Explain.

g) recall that the wavefunction for this particle after it has been scattered may be written $psi_{sc}=Nf(theta)(e^{ikr}/r)$, for large values of the radial distance from the scattering center $r$.

i) Write the scattered wavefunction for this particle in terms of a superposition of $|klm>$. Do the coefficients in your expression depend on $k$,$l$ or $m$? Explain.

Consider a measurement of the orbital angular moment squared (corresponding to the angular moment $l$ of both the incident and scaterred particle. 

ii) Is the probability that a given value of $l$ is measured for the scattered particle greated than, less than or equal to the probability that the same value of $l$ is measured for the incident particle? Explain. 

iii) use the answer to the previous question to relate the coefficients in your expression for the incident and scattered wavefunction. Explain.

The angular momentum is simply

$|L ⃗|=|r ⃗×p ⃗|=rmv*sin⁡ α=b*mv$

$L ⃗$  is perpendicular to the paper into the paper so that $L_z=0$

b,c) In all collisions angular momentum is conserved, so $L ⃗$ stays the same after collision and also $L_z$. (After collision direction is into the paper, $Lz=0$)

d, e) $|L ⃗ |=b*mv$ so that if $b$ increases $|L ⃗|$ increases, but the direction stays the same (into the paper)

$ψ_{incident} (r,θ,φ)=Ne^{ikz}$    (a plane wave in the z direction)
f)

-momentum is $p=ℏk$ so is well defined since k is given directly in the expression of $ψ$.

– $L_z=x*p_y-y*p_x$    and since $[x_i,p_j ]=iℏδ_{ij} ? 0$ for $i=j$ it means one cannot know simultaneously $x$ and $px$ or $y$ and $py$. So that $L_z$  is not well defined.
–  since $[L^2,L_z ]=0$ they share a common set of eigenvectors. Since $L_z$ is not well defined, neither is $L^2$.

– The expansion of a plane wave into spherical waves is given by equation [11.28] (Rayleigh):

$e^{ikz}=∑_(l=0)^∞i^l  (2l+1)*j_l (kr) P_l (cos⁡(θ))$

We know $Y_l^0=sqrt{((2l+1)/4π)}*P_l (cos⁡ θ)$ so that

$e^{ikz}=∑_l i^l sqrt{(4π(2l+1))}*j_l (kr)*Y_l^0 (θ,φ)=∑_lA_{kl}*|k,l,0>$

with $A_{kl}=i^l sqrt{(4π(2l+1))}*j_l (kr)$  depending on $k$ and $l$ only

j_l (kr)  are called the Bessel functions

g)

i)

The scattered wave is

$ψ_{scattered} (r,θ,φ)=N*f(θ)*e^{ikr/r}$

And can be written as the expansion:

$ψ_{sc}=k∑_l i^(l+1) (2l+1) *a_l*h_l (kr)*P_l (cos⁡ θ)$

Since $Y_l^m=sqrt{((2l+1)/4π)}*P_l (cos⁡ θ)$ it follows that

$ψ_{sc}=∑_l ki^(l+1) sqrt{(4π(2l+1))}*a_l*h_l (kr)*Y_l^0 (θ,φ)=$

$= ∑_l B_kl*|k,l,0>$

Where

$B_{kl}=ki^(l+1) sqrt{(4π(2l+1))}*a_l*h_l (kr)$  are the coefficients depending on $k$ and $l$ only

$h_l (kr)$  are called Hankel functions and $a_(l )$ are numbers

ii) Since $L ⃗$ is conserved in a scattering then before and after scattering the probability to measure the value l (quantum numebr of $L^2$) is the same.

iii)

Since the incident and scattered wave can be expanded as a sum of the same (eigen)vectors $|k,l,0>$, the probabilities of measuring the value $l$ (for $L^2$) in the incident and scattered waves are

$P_{(l-incident)}=|A_{kl} |^2$     and $P_{(l-scattered)}=|B_{kl} |^2$
As shown in ii) these two probabilities are equal so that

$i^l sqrt{(4π(2l+1))}*j_l (kr)=(±)ki^{(l+1)} sqrt{(4π(2l+1))}*a_l*h_l (kr)$

$j_l (kr)=(±)ki*a_l*h_l (kr)$

In fact the definition of Hankel functions is

$h_l (x)=j_l (x)±i*n_l (x)$   where $n_l=(j_l*cos⁡ (lπ)-j_{(-l)} )/sin⁡(lπ)$

So the above equation holds.