EE201/MSE207 Lecture 11 Electron spin in magnetic field 휇 Interaction energy 퐻 = −휇 퐵 is magnetic dipole moment (as for compass needle) 퐵 is magnetic field, dot-product

휇 = 훾orbital 퐿 or 휇 = 훾spin 푆 훾 is called gyromagnetic ratio

In classical physics: 푞 charge (−푒 for electron) 훾cl = 2푚 mass In quantum mechanics:

For orbital motion 훾orbital = 훾푐푙

For spin 훾spin = 푔 훾푐푙 g-factor 푔 ≈ 2.0023 for free electron, different values in materials In more detail 푚 Angular momentum 휔 푞 Classical physics 푅 퐿 = 푚푣푅 = 푚푅2휔 Magnetic moment 푞 휔 휇 = 퐼퐴 = 휋푅2 = 푞 휋푅2 휇 푞 current 푇 2휋 훾 = = area Therefore cl 퐿 2푚 Quantum, orbital motion 푒ℏ The same 훾, 퐿 = 푛ℏ  휇 = −푛 푧 푧 2푚 magnetic quantum number (usually 푚) The smallest value (quantum 푒ℏ of magnetic moment) 휇 = (Bohr magneton) 퐵 2푚 Quantum, spin 푆푧 = ±ℏ/2, but still 휇푧 ≈ ±휇퐵 (a little more, 휇푧 ≈ ±1.001 휇퐵 ), so 훾spin ≈ 2훾cl (i.e., 푔 ≈ 2). (Different people use different notations; sometimes 푔 = −2, sometimes 훾 is positive, sometimes 푔 is called gyromagnetic ratio, etc.) Evolution of spin in magnetic field 퐵 푑퐿 퐻 = −휇 퐵 = −훾퐵푆 torque 휇 × 퐵 = 훾퐿 × 퐵 휇 푑푡 Classically, precession 푑퐿푥 푑퐿푦 = 훾퐿푦퐵푧 = −훾퐿 퐵 as in a gyroscope. 푑푡 푑푡 푥 푧 (assume 퐵 = 퐵푧푘) Larmor precession, 휔 = 훾퐵. 2 푑 퐿푥 = −훾2퐵2퐿 ⇒ |휔| = |훾퐵 | 푑푡2 푧 푥 푧 퐵 Quantum mechanics ℏ 1 0 Assume 퐵 = 퐵 푘 퐻 = −훾퐵푆 = −훾퐵 푧 2 0 −1 푑휒 푑휒 푖 훾 is negative, −훾 is positive 푖ℏ = 퐻 휒 ⇒ = − 퐻 휒 푑푡 푑푡 ℏ 푑푎 푡 푖훾퐵 = 푎 푡 푎 푡 푎 0 푒푖훾퐵푡/2 휒 푡 = ⇒ 푑푡 2 ⇒ 휒 푡 = 푏 푡 푑푏 푡 푖훾퐵 푏 0 푒−푖훾퐵푡/2 = − 푏 푡 푑푡 2 0 1 푎 2 + 푏 2 = 1 Both and are eigenvectors 1 0 Average values 푖훾퐵푡/2 푎 = 푎(0) 휒 푡 = 푎 푒 푏 푒−푖훾퐵푡/2 푏 = 푏(0)

푖훾퐵푡 2 2 2 Obviously 푆푍 = const, since 푎푒 = 푎 and also precession about z-axis. Nevertheless, let us check formally. ℏ 1 0 푎 푒푖훾퐵푡/2 푆 푡 = 휒 푆 휒 = 푎∗푒−푖훾퐵푡/2, 푏∗푒푖훾퐵푡/2 = 푧 푧 2 0 −1 푏 푒−푖훾퐵푡/2 ℏ = ( 푎 2 − 푏 2) Yes, does not depend on 푡. 2 Similarly ℏ 0 1 푎 푒푖훾퐵푡/2 푆 푡 = 휒 푆 휒 = 푎∗푒−푖훾퐵푡/2, 푏∗푒푖훾퐵푡/2 = 푥 푥 2 1 0 푏 푒−푖훾퐵푡/2 ℏ = 푎∗푏 푒−푖훾퐵푡 + 푎푏∗푒푖훾퐵푡 = ℏ Re(푎∗푏 푒−푖훾퐵푡) 2 oscillations with frequency 휔 = 훾퐵 ℏ 0 −푖 푎 푒푖훾퐵푡/2 푆 푡 = 푎∗푒−푖훾퐵푡/2, 푏∗푒푖훾퐵푡/2 = ℏ Im(푎∗푏푒−푖훾퐵푡) 푦 2 푖 0 푏 푒−푖훾퐵푡/2 Dynamics of average values

ℏ 2 2 ∗ −푖훾퐵푡 ∗ −푖훾퐵푡 푆 푡 = 푎 − 푏 , 푆 푡 = ℏ Re 푎 푏 푒 , 푆푦 푡 = ℏ Im(푎 푏푒 ) 푧 2 푥

If 푎 = cos(훼 2) then ℏ 푆 푡 = sin 훼 cos(훾퐵푡) 푏 = sin(훼 2) 푥 2 ℏ 푆 푡 = − sin 훼 sin(훾퐵푡) 푦 2 ℏ 푆 푡 = cos 훼 푧 2 Dynamics of 〈푆 푡 〉 is the same as in the classical case, 휔 = 훾퐵.

A way to visualize (as if) 퐵 However, remember that actually 3 푆2 = ℏ ≈ 0.87 ℏ 훼 4 ℏ and if we measure 푆 , 푆 , or 푆 , we always get ± . 푥 푦 푧 2 Measurement 푖훾퐵푡/2 휒 푡 = 푎 푒 푏 푒−푖훾퐵푡/2 푖훾퐵푡/2 2 2 If we measure 푆푧, then we get + ℏ 2 with probability 푎 푒 = 푎 .

If we measure 푆푥, then we get + ℏ 2 with probability 푖훾퐵푡 2 1 1 2 푎 푒 2 푃 푆푥 = + ℏ 2 = 〈휒푥+|휒〉 = , 푖훾퐵푡 = 2 2 − 푏 푒 2 1 2 1 2 = 푎푒푖훾퐵푡/2 + 푏 푒−푖훾퐵푡/2 = 푎 + 푏 푒−푖훾퐵푡 2 2 1 2 Similarly, 푃 푆 = − ℏ 2 = 푎 − 푏 푒−푖훾퐵푡 frequency 휔 = 훾퐵 푥 2 Special case: 푎 = 푏 = 1 2, then 1 1 푃 푆 = + ℏ 2 = cos 훾퐵푡 2 = + cos(훾퐵푡) (in general, amplitude 푥 2 2 1 1 can be smaller and also 푃 푆 = − ℏ 2 = sin 훾퐵푡 2 = − cos 훾퐵푡 푥 2 2 extra phase shift) Quantum coherent oscillations: oscillations of probability (if measured)

If we measure 푆푥, the state will be collapsed onto 푥-axis (± ℏ 2), 푆푧 and 푆푦 components will be lost. Another way to consider evolution (in 푥-basis) (not included into this course) 푧-basis ⟶ 푥-basis

ℏ 1 0 ℏ 0 1 퐻 = −훾퐵 ⟶ −훾퐵 2 0 −1 푧 2 1 0 (푥) (Rabi oscillations in a qubit) 푑휒 푎푥 휒푥 = 푖ℏ = 퐻휒 푏푥 푑푡 푑푎 ℏ 푖 ℏ 푥 = −훾퐵 푏 푑푡 2 푥  oscillations with frequency 훾퐵/2, 푑푏 ℏ 푖 ℏ 푥 = −훾퐵 푎 probabilities will oscillate with freq. 훾퐵 푑푡 2 푥 (not included into this course) Experimental measurement of spin (Stern-Gerlach experiment, 1922)

N inhomogeneous magnetic field produces force onto a magnetic moment

neutral 퐻 = −훾퐵푆 atoms angular momentum (or spin) S magnetic field gyromagnetic ratio If 퐵 is inhomogeneous (not constant), then 퐹 = −훻퐻 = 훾훻(퐵푆 ) So, force depends on 푆 . 휕퐵 휕퐵 휕퐵 If 퐵 푥, 푦, 푧 = 퐵 + 훼푧 푘 − 훼푥푖 (for magnetic field 푥 + 푦 + 푧 = 0), 0 휕푥 휕푦 휕푧 not important then 퐹 = 훾훼 푆푧푘 − 푆푥푖 not important (oscillates because of Larmor precession about z-axis, so zero on average) 퐹푧 = 훾훼푆푧 spin-up is deflected down (훾 < 0), (particle should be neutral because otherwise spin-down is deflected up charge will circle in magnetic field) Addition of spins (similar for angular momenta)

(1) (2) (1) (2) Two particles 푆 = 푆 + 푆 푆푧 = 푆푧 + 푆푧 (vectors added) (scalars added) 2 2 However, not simple for the total spin 푆2 = 푆 1 + 푆 2 + 2 푆 (1)푆 (2)

Two particles with spin 1/2 훼↑↑ 훼 휒 = 훼 ↑↑ + 훼 ↑↓ + 훼 ↓↑ + 훼 ↓↓ = ↑↓ ↑↑ ↑↓ ↓↑ ↓↓ 훼↓↑ 1 1 훼↓↓ ↑↑ 푚 = 2 + 2 = 1 (푆푧= 1 ∙ ℏ), 푠 = 1 ↑↓ 푚 = 0 these states are not eigenstates of the total spin 푆2 ↓↑ 푚 = 0 1 1 ↓↓ 푚 = − 2 − 2 = −1 (푆푧= −1 ∙ ℏ), s = 1 Eigenstates ( ↑↓ − ↓↑ )/ 2 푠 = 0 (called singlet), 푚 = 0 of total spin: ↑↑ 푚 = 1 ( ↑↓ + ↓↑ )/ 2 푠 = 1 푚 = 0 (easy to check) (triplet) ↓↓ 푚 = −1 Addition of arbitrary spins (or angular momenta)

Addition of arbitrary spins 푠1 and 푠2 is quite complicated.

Possible values range from 푠1 + 푠2 to |푠1 − 푠2| (integer ladder).

Eigenvectors are given by Clebsch-Gordan coefficients.

Example quarks: 푠 = 1 2 two quarks: 1 2 + 1 2 = 1 or 0 (mesons: vector and pseudoscalar) three quarks: 1 or 0 + 1 2 = 1 2 or 3 2

proton, Delta, neutron, Omega, etc. etc.