- 1 A Consider The Standard Matrix Representation Of The Spin 1 2 Operators 5x 2 6 5 1 5 Sz 1 69 Ident 1 (131.74 KiB) Viewed 50 times
1. (a) Consider the standard matrix representation of the spin-1/2 operators 5x = ²2 (²6) 5 = ² (1 5¹), sz=1/(69). Identify the basis being used, explaining your reasoning. State the physical interpretation of the components of the spinor in this basis. [3 marks] (b) A spinor for spin-1/2 may be written in the form (cos(0/2), e* sin(0/2)). Obtain the corresponding spin-vector, i.e., the vector of expectation values of the spin operators. Describe this spin vector using words or a diagram, stating how it is related to the angles 8 and 4. [5 marks] (c) The Hamiltonian for an electron spin in a magnetic field is H = (gμB/h) ³. B. Suppose that the field is along the z axis and independent of time. Using Schrodinger's equation, or otherwise, obtain the spinor at time t in terms of its value at time 0. By comparing your result with the form in part (b), show that the spin-vector precesses around the magnetic field, and obtain the frequency of this precession. [5 marks] (d) At time t=0 the electron spin was measured along x and found to be ħ/2. The electron subsequently experiences a magnetic field along z. Draw the motion of the spin vector for times > 0. Using your diagram, or otherwise, deduce those times, if any, for which there is no uncertainty in the x, y, and z spin components, explaining your reasoning. [5 marks] (e) Suppose the electron moves in a field including both constant and oscillating parts, B = (Bo cos(wt), Bo sin(wr), B₂). Consider two different ways to write the spinor, (a(t), b(t)) = (A(1), B(t)e), where z is a constant. Write down the differential equations for the functions a(t), b(t), A(t) and B(1). Deduce that for certain values of w, the spin-vector corresponding to a spinor (A(1), B(t)) precesses around a time-independent magnetic field. Give the value or values of w for which this occurs, and briefly describe the associated motion of the spin-vector, If, in this situation, the spin was down at time 0, at what subsequent time or times will it be up? [7 marks]