Electron spin in magnetic field is magnetic dipole moment - - PowerPoint PPT Presentation

EE201/MSE207 Lecture 11

Electron spin in magnetic field

charge (−𝑓 for electron)

𝐼 = − 𝜈𝐶

In classical physics: Interaction energy (as for compass needle) 𝜈 is magnetic dipole moment 𝐶 is magnetic field, dot-product

𝜈 = 𝛿orbital 𝑀 or 𝜈 = 𝛿spin 𝑇

𝛿 is called gyromagnetic ratio

𝛿cl = 𝑟 2𝑛

mass In quantum mechanics: For orbital motion 𝛿orbital = 𝛿𝑑𝑚 For spin 𝛿spin = 𝑕 𝛿𝑑𝑚 g-factor

𝑕 ≈ 2.0023 for free electron,

different values in materials

In more detail Classical physics

𝑆

𝑛 𝑟 𝜕 Angular momentum

𝑀 = 𝑛𝑤𝑆 = 𝑛𝑆2𝜕

Magnetic moment

𝜈 = 𝐽𝐵 = 𝑟 𝑈 𝜌𝑆2 = 𝑟 𝜕 2𝜌 𝜌𝑆2

current area Therefore

𝛿cl = 𝜈 𝑀 = 𝑟 2𝑛

Quantum, orbital motion The same 𝛿, 𝑀𝑨 = 𝑜ℏ 

𝜈𝑨 = −𝑜 𝑓ℏ 2𝑛

magnetic quantum number (usually 𝑛) The smallest value (quantum

• f magnetic moment)

𝜈𝐶 = 𝑓ℏ 2𝑛

(Bohr magneton) Quantum, spin

𝑇𝑨 = ±ℏ/2, but still 𝜈𝑨 ≈ ±𝜈𝐶 (a little more, 𝜈𝑨 ≈ ±1.001 𝜈𝐶 ),

so 𝛿spin ≈ 2𝛿cl (i.e., 𝑕 ≈ 2).

(Different people use different notations; sometimes 𝑕 = −2, sometimes 𝛿 is positive, sometimes 𝑕 is called gyromagnetic ratio, etc.)

Evolution of spin in magnetic field

𝐼 = − 𝜈𝐶 = −𝛿𝐶 𝑇

𝐶 𝜈 Classically, precession as in a gyroscope. Larmor precession, 𝜕 = 𝛿𝐶.

torque

𝜈 × 𝐶 𝑒𝑀 𝑒𝑢 = 𝛿𝑀 × 𝐶 𝑒𝑀𝑦 𝑒𝑢 = 𝛿𝑀𝑧𝐶𝑨 𝑒𝑀𝑧 𝑒𝑢 = −𝛿𝑀𝑦𝐶𝑨

(assume 𝐶 = 𝐶𝑨𝑙)

𝑒2𝑀𝑦 𝑒𝑢2 = −𝛿2𝐶𝑨

2𝑀𝑦 ⇒ |𝜕| = |𝛿𝐶𝑨|

Quantum mechanics Assume 𝐶 = 𝐶 𝑙 𝐶

𝐼 = −𝛿𝐶 𝑇𝑨 = −𝛿𝐶 ℏ 2 1 −1

𝛿 is negative, −𝛿 is positive

𝑗ℏ 𝑒𝜓 𝑒𝑢 = 𝐼𝜓 ⇒ 𝑒𝜓 𝑒𝑢 = − 𝑗 ℏ 𝐼𝜓 𝜓 𝑢 = 𝑏 𝑢 𝑐 𝑢 ⇒ 𝑒𝑏 𝑢 𝑒𝑢 = 𝑗𝛿𝐶 2 𝑏 𝑢 𝑒𝑐 𝑢 𝑒𝑢 = − 𝑗𝛿𝐶 2 𝑐 𝑢 ⇒ 𝜓 𝑢 = 𝑏 0 𝑓𝑗𝛿𝐶𝑢/2 𝑐 0 𝑓−𝑗𝛿𝐶𝑢/2 𝑏 2 + 𝑐 2 = 1

Both 0 1 and 1 0 are eigenvectors

Average values

𝜓 𝑢 = 𝑏 𝑓𝑗𝛿𝐶𝑢/2 𝑐 𝑓−𝑗𝛿𝐶𝑢/2

Obviously 𝑇𝑎 = const, since 𝑏𝑓

𝑗𝛿𝐶𝑢 2 2 = 𝑏 2 and also precession about z-axis.

𝑏 = 𝑏(0) 𝑐 = 𝑐(0)

Nevertheless, let us check formally.

𝑇𝑨 𝑢 = 𝜓 𝑇𝑨 𝜓 = 𝑏∗𝑓−𝑗𝛿𝐶𝑢/2, 𝑐∗𝑓𝑗𝛿𝐶𝑢/2 ℏ 2 1 −1 𝑏 𝑓𝑗𝛿𝐶𝑢/2 𝑐 𝑓−𝑗𝛿𝐶𝑢/2 = = ℏ 2 ( 𝑏 2 − 𝑐 2)

Yes, does not depend on 𝑢. Similarly

𝑇𝑦 𝑢 = 𝜓 𝑇𝑦 𝜓 = 𝑏∗𝑓−𝑗𝛿𝐶𝑢/2, 𝑐∗𝑓𝑗𝛿𝐶𝑢/2 ℏ 2 1 1 𝑏 𝑓𝑗𝛿𝐶𝑢/2 𝑐 𝑓−𝑗𝛿𝐶𝑢/2 = = ℏ 2 𝑏∗𝑐 𝑓−𝑗𝛿𝐶𝑢 + 𝑏𝑐∗𝑓𝑗𝛿𝐶𝑢 = ℏ Re(𝑏∗𝑐 𝑓−𝑗𝛿𝐶𝑢)

• scillations with frequency 𝜕 = 𝛿𝐶

𝑇𝑧 𝑢 = 𝑏∗𝑓−𝑗𝛿𝐶𝑢/2, 𝑐∗𝑓𝑗𝛿𝐶𝑢/2 ℏ 2 −𝑗 𝑗 𝑏 𝑓𝑗𝛿𝐶𝑢/2 𝑐 𝑓−𝑗𝛿𝐶𝑢/2 = ℏ Im(𝑏∗𝑐𝑓−𝑗𝛿𝐶𝑢)

Dynamics of average values

𝑇𝑨 𝑢 = ℏ 2 𝑏 2 − 𝑐 2 , 𝑇𝑦 𝑢 = ℏ Re 𝑏∗𝑐 𝑓−𝑗𝛿𝐶𝑢 , 𝑇𝑧 𝑢 = ℏ Im(𝑏∗𝑐𝑓−𝑗𝛿𝐶𝑢) If 𝑏 = cos( 𝛽 2) 𝑐 = sin( 𝛽 2) then

𝑇𝑦 𝑢 = ℏ 2 sin 𝛽 cos(𝛿𝐶𝑢) 𝑇𝑧 𝑢 = − ℏ 2 sin 𝛽 sin(𝛿𝐶𝑢) 𝑇𝑨 𝑢 = ℏ 2 cos 𝛽

Dynamics of 〈 𝑇 𝑢 〉 is the same as in the classical case, 𝜕 = 𝛿𝐶.

𝛽

𝐶 A way to visualize (as if) However, remember that actually 𝑇2 = 3 4 ℏ ≈ 0.87 ℏ and if we measure 𝑇𝑦, 𝑇𝑧, or 𝑇𝑨, we always get ±

ℏ 2.

Measurement

𝜓 𝑢 = 𝑏 𝑓𝑗𝛿𝐶𝑢/2 𝑐 𝑓−𝑗𝛿𝐶𝑢/2

If we measure 𝑇𝑨, then we get + ℏ 2 with probability 𝑏 𝑓𝑗𝛿𝐶𝑢/2 2 = 𝑏 2. If we measure 𝑇𝑦, then we get +

ℏ 2 with probability

𝑄 𝑇𝑦 = + ℏ 2 = 〈𝜓𝑦+|𝜓〉 2 = 1 2 , 1 2 𝑏 𝑓

𝑗𝛿𝐶𝑢 2

𝑐 𝑓−𝑗𝛿𝐶𝑢

2 2

= = 1 2 𝑏𝑓𝑗𝛿𝐶𝑢/2 + 𝑐 𝑓−𝑗𝛿𝐶𝑢/2 2 = 1 2 𝑏 + 𝑐 𝑓−𝑗𝛿𝐶𝑢 2 Similarly,

𝑄 𝑇𝑦 = − ℏ 2 = 1 2 𝑏 − 𝑐 𝑓−𝑗𝛿𝐶𝑢 2

frequency 𝜕 = 𝛿𝐶 Special case: 𝑏 = 𝑐 = 1 2, then 𝑄 𝑇𝑦 = + ℏ 2 = cos 𝛿𝐶𝑢 2 =

1 2 + 1 2 cos(𝛿𝐶𝑢)

𝑄 𝑇𝑦 = − ℏ 2 = sin 𝛿𝐶𝑢 2 =

1 2 − 1 2 cos 𝛿𝐶𝑢

(in general, amplitude can be smaller and also extra phase shift)

Quantum coherent oscillations: oscillations of probability (if measured) If we measure 𝑇𝑦, the state will be collapsed onto 𝑦-axis (± ℏ 2), 𝑇𝑨 and 𝑇𝑧 components will be lost.

Another way to consider evolution (in 𝑦-basis)

(not included into this course)

𝑨-basis ⟶ 𝑦-basis 𝐼 = −𝛿𝐶 ℏ 2 1 −1

𝑨

⟶ −𝛿𝐶 ℏ 2 1 1 0 (𝑦)

(Rabi oscillations in a qubit)

𝜓𝑦 = 𝑏𝑦 𝑐𝑦 𝑗 ℏ 𝑒𝑏𝑦 𝑒𝑢 = −𝛿𝐶 ℏ 2 𝑐𝑦 𝑗ℏ 𝑒𝜓 𝑒𝑢 = 𝐼𝜓 𝑗 ℏ 𝑒𝑐𝑦 𝑒𝑢 = −𝛿𝐶 ℏ 2 𝑏𝑦



• scillations with frequency 𝛿𝐶/2,

probabilities will oscillate with freq. 𝛿𝐶

Experimental measurement of spin (Stern-Gerlach experiment, 1922)

N S

neutral atoms

inhomogeneous magnetic field produces force onto a magnetic moment

𝐼 = −𝛿𝐶 𝑇

magnetic field angular momentum (or spin) gyromagnetic ratio

If 𝐶 is inhomogeneous (not constant), then

𝐺 = −𝛼𝐼 = 𝛿𝛼(𝐶 𝑇)

So, force depends on 𝑇. If 𝐶 𝑦, 𝑧, 𝑨 = 𝐶0 + 𝛽𝑨 𝑙 − 𝛽𝑦

𝑗

(for magnetic field

𝜖𝐶𝑦 𝜖𝑦 + 𝜖𝐶𝑧 𝜖𝑧 + 𝜖𝐶𝑨 𝜖𝑨 = 0),

then

𝐺 = 𝛿𝛽 𝑇𝑨𝑙 − 𝑇𝑦 𝑗

not important not important (oscillates because of Larmor precession about z-axis, so zero on average)

𝐺

𝑨 = 𝛿𝛽𝑇𝑨

spin-up is deflected down (𝛿 < 0), spin-down is deflected up

(particle should be neutral because otherwise charge will circle in magnetic field)

(not included into this course)

Addition of spins (similar for angular momenta)

Two particles

𝑇 = 𝑇(1) + 𝑇(2) 𝑇𝑨 = 𝑇𝑨

(1) + 𝑇𝑨 (2)

(vectors added) (scalars added)

However, not simple for the total spin

𝑇2 = 𝑇 1

2 + 𝑇 2 2 + 2

𝑇(1) 𝑇(2)

Two particles with spin 1/2

𝜓 = 𝛽↑↑ ↑↑ + 𝛽↑↓ ↑↓ + 𝛽↓↑ ↓↑ + 𝛽↓↓ ↓↓ = 𝛽↑↑ 𝛽↑↓ 𝛽↓↑ 𝛽↓↓ ↑↑ ↑↓ ↓↑ ↓↓

𝑛 =

1 2 + 1 2 = 1 (𝑇𝑨= 1 ∙ ℏ), 𝑡 = 1

𝑛 = −

1 2 − 1 2 = −1 (𝑇𝑨= −1 ∙ ℏ), s = 1

these states are not eigenstates of the total spin 𝑇2 𝑛 = 0 𝑛 = 0 Eigenstates

• f total spin:

( ↑↓ − ↓↑ )/ 2 ↑↑ ( ↑↓ + ↓↑ )/ 2 ↓↓

𝑡 = 0 (called singlet), 𝑛 = 0 𝑡 = 1 (triplet) 𝑛 = 1 𝑛 = 0 𝑛 = −1

(easy to check)

Addition of arbitrary spins (or angular momenta)

Addition of arbitrary spins 𝑡1 and 𝑡2 is quite complicated. Possible values range from 𝑡1 + 𝑡2 to |𝑡1 − 𝑡2| (integer ladder). Eigenvectors are given by Clebsch-Gordan coefficients. Example quarks: 𝑡 = 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, neutron, etc. Delta, Omega, etc.