**Problem 1**

Solution 1

At Fermilab anti-protons p' are produced by colliding a beam of protons p with protons at rest in the reaction p+p → p+p+p+ p'. In such a “fixed target experiment” the center-of-mass system is defined as the reference frame where the total momentum

**p**

_{CM}of the system is zero.

(a) Show that for a particle of mass m incident with kinetic energy T

_{L}(in the lab frame) on a target of mass M, the total center-of-mass energy E

_{CM}is given by E

^{2}

_{CM}= (M+m)

^{2}+2MT

_{L}.

(b) What minimum kinetic beam energy T

_{B}is required to produce anti-protons with mass m

_{p}= 938 MeV in the process described above.

(c) In the actual experiment at Fermilab, the protons are collided with a block of heavy metal. This means the protons in the target are bound inside heavy nuclei with the result that the protons have a kinetic energy T

_{T}. What incident beam energy E

_{B}would be required to produce anti-protons off of those protons with T

_{T}, if their velocity was directed toward the beam. Express your answer in terms of E

_{T}, p

_{T}and m

_{p}.

(d) Suppose the kinetic energy T

_{T}of such protons inside heavy nuclei is 40 MeV. What is the velocity β of these protons.

**Problem 2**

Solution 2

Suppose coordinate system K′ is moving with respect to coordinate system K with velocity

**K**in arbitrary direction. Determine the Lorentz transformation which connects the coordinates (ct′, x′, y′, z′) in K′ to the coordinates (ct, x, y, z) in K.

**Problem 3**

Solution 3

A cone-shaped region of angle α is obtained from a circle with radius a. Consider the electrostatic potential in this two-dimensional region where the potential φ is zero along the edges (the two radial parts and the arc of the circle). Find the Green’s function for this boundary value problem. Note that the two-dimensional Green’s function is defined by

∇

^{2}G(

**x**,

**x**′) = δ

^{2}(

**x**−

**x**′)

with

∇

^{2}= ∂

^{2}/∂x

^{2}+ ∂

^{2}/∂y

^{2}∂2 = (1/r)(∂/∂r)[r(∂/∂r)] + (1/r

^{2})(∂

^{2}/∂θ

^{2})

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