HW- s= et -3 (spin 1/2 and denstiy operato= r)

Due date: Feb. 5, 2009

3.1. Use the commutation relation of angular momentum operators, i.e., follow p171-174 of Griffths or Merzbacher p.238 for the sp= in 1/2 systems to derive the mat= rix representation of the Pauli matrices as shown in eq. (1) on note page 4.2.

3.2. Derive the four equations under Eq. (2) on the sa= me page. For the last equation, find the coefficients for a matrix where A11=3D1+i , A₁₂=3D1+2i; A₂₁=3D1.0 and A₂₂=3D2+i.

3.3. Sho= w that

= =

where is a unit vector. "Remember" this equati= on which will be used often.

3.4. This is for a spin 1/2 system. The matrix representation of any operator that is defined with respect to the base sta= tes which are eigenstates of is called standard representation. All the matrice below will be referred to the standard representation. Let be t= he polar angles.

(a) = Write down in t= erms of x,y, z components of Writ= e down the matrix representation of in t= he standard base.

(b) = Show that given on page 4.3 are indeed the eigenstates of .

(c) = If you measure S_zfor a system of particles in the pure state , what values will you get each time, and what is = the average value you will get?

(d) = Suppose that there are 10⁶ particles in part (c) to start with, write do= wn the matrix representation of the state operator (or density operator) before the measurement and after the measurement.

(e)<= span style=3D'mso-spacerun:yes'> If 30% of the 10⁶ parti= cles are in the stat= e, and 70% are in the stat= e. What is the expectation value of S_z for such an ensemble. Use the intuitive approach.

We wi= ll next solve the same problem in (e) using state operator below.

(f) = Write down the state operator in the operator form.

(g) = Express the same operator in matrix form.

(h) Calculate the expectation value of S_z and compare with your resu= lts from (e).

(i)<= span style=3D'mso-spacerun:yes'> Calculate the expectation value of= S_y.

3.5.= Do the exercises shown on Pages 4.= 6, 4.7, 4.9.

Study Aid—

· Wikipedia has a good short introduction on electron spin. See

= &nb= sp; http://= www.answers.com/topic/spin-quantum-number

Pick up a few phases while = you are there: Levi-Civita symbols; Uhlenbeck and Goudsmit, = Ehrenfest, Kronig, Lande . Bohr magneton. Skip the Dirac equation.

· This link goes to a higher level which is not needed for us at this point, but you might want to take a quick look http://www.answers.com/topic/spi= n

· You may want to learn a bit about isospin while we are talking about spin. This part below is adopted from: http://www.answers.com/isospin

Isospin was introduced by Wer= ner Heisenberg in 1932^[1] (although it was named by Eugene Wigner in 1937^[2]) to explain symmetries of the then newly discovered neutron:<= /p>

The mass of= the neutron and the proton are almost identical: they are nearly degenerat= e, and both are thus often called nucleons= . Although the proton has a positive charge, and the neutron is neutral,= they are almost identical in all other respects.
The strength of the strong interaction between any pair of nucleons is the same, independent of whether they are interacting as protons or as neutrons.

Thus, isospin was introduced as a concept well before the development in= the 1960s of the quark model which provides our modern understanding.

The nuc= leons, baryons of spin ¹⁄₂₌, were grouped together because they both have nearly the same mass and inter= act in nearly the same way. Thus, it was convenient to treat them as being different states of the same particle. Since a spin ¹⁄₂₌ particle has two states, the two were said to be of isospin ¹⁄₂. The proton and neutron were then associated with different isospin projections I_z<=
/sub> =3D +¹⁄₂ and −¹⁄₂ respectively. = When constructing a physical theory of nuclear= forces, one could then simply assume that it does not depend on isospin.

These considerations would also prove useful in the analysis of meson-nucle= on interactions after the discovery of the pions in 194= 7. The three pions (π⁺, π⁰, π= ;⁻) could be assigned to an isospin triplet with I =3D 1 and <= i>I_z =3D +1, 0 or −1. By assuming that isospin was conserved by nuclear interactio= ns, the new mesons were more easily accommodated by nuclear theory.

As further particles were discovered, they were assigned into isospin multiplets according to the number of different charge states seen: a doubl= et I =3D ¹⁄₂ of K me= sons, a triplet I =3D 1 of Σ baryons, a single I&nbs= p;=3D 0 Λ, four I =3D ³⁄₂₌ Δ baryons, and so on. This multiplet structure was combined with strangeness= in Murr= ay Gell-Mann's Eightfold way (physics), ultimately leading to the quark mod= el and quantum chromodynamics.