Question

3. (a) A particle orbits a fixed point in space. Given that L-=-ih- , find the eigenvalues аф and normalised eigenfunctions of the z-component of the particle's angular momentum. (5 marks) Explain clearly your reasoning at each important step An electron is in the spin state x)-A G) Normalise x) and hence determine the constant A (ii) If you measured the z-component of the electron's spin, s, . what values could (b) in the usual s basis you get and what is the probability of obtaining each? What is the expectation value of s,? (iii) If, instead, you measured the x-component of the electron's spin, s, , what values could you get and what is the probability of obtaining each? expectation value of s,? What is the The Pauli Matrices are: σ,- 5 marks)

Answer 1

Given L_2 = -i planckconstantover2pi partial differential/partial differential phi = m planckconstantover2pi L_z psi(phi) = -i planckconstantover d psi(phi)/d phi = m plankconstantover2pi psi(phi) = d psi(phi)/psi(phi) = in d phi integrating on both sides log psi(phi) = im phi + log alpha psi Vertical-bar phi Vertical-bar /alpha = e^i m phi psi Vertical-bar phi Vertical-bar = alpha e^im phi therefore 0 lessthanorequalto phi lessthanorequalto 2 pi so Normalization 1 = integral^2pi_0 psi^* Vertical-bar phi Vertical-bar psi Vertical-bar phi Vertical-bar d phi 1 = Vertical-bar alpha Vertical-bar ^2 integral^2pi_0 e^im phi e^im phi d phi = Vertical-bar alpha Vertical-bar ^2 times Integral^2pi_0 d phi 1 = Vertical-bar alpha Vertical-bar ^2 times 2 pi alpha = 1/squareroot 2 pi psi(phi) = 1/ squareroot 2 pi e^im phi (b) Vertical-bar + >_z = (1 0) , & 1- >_z = (0 1) are the basic of s_z Vertical-bar x > = A(1 - 2i 2) = A[(1 - 2i) (1 0) + 2(0 1)] Vertical-bar x > = A [1 - 2 i Vertical-bar + >_z + delta Vertical-bar - >_z Vertical-bar ] (i) normalization 1 = < x Vertical-bar x > = Vertical-bar A Vertical-bar ^2 [(1 + 2 i) (1 - 2 i)_2