The Physics of Spin Otto Stern Walther Gerlach 1922: Wrong theory  right experiment Walther Gerlach

= N S Deflected Up Undeflected Beam of Atoms Oven Stern-Gerlach Detection Screen = Tiny Spinning Bar Magnet Silver Atom

= STONGER Nonuniform Magnetic Field N S N S N S N S N S Deflected Up N weaker N S STONGER Deflected Up

= STONGER Nonuniform Magnetic Field N S S N S N N S Undeflected N S weaker N S N S N S STONGER Undeflected S

This result has no “classical” or “commonsense” explanation Experiment #1 50% Random Spin Directions 50% x y z ACTUALLY SEE (!!) EXPECT TO SEE This result has no “classical” or “commonsense” explanation

Experiment #2 100% 0% 50% 1 “Working Hypothesis” 2 50% 0% 100% Random 50% 0% 100% It is consistent to hypothesize: Beam 1 atoms are all “spin up”: Beam 2 atoms are all “spin down”:

Experiment #3 z y x 50% 50% 50% HORIZONTAL VERTICAL 50% Random 50% 50% HORIZONTAL VERTICAL x y z 50% HORIZONTAL atom has 50/50 CHANCE of emerging “spin up” or “spin down”

Summary of Experiments 100% 0% 100% 0% 50% 50% 1: “Quantization” of spin direction 2: Probability plays fundamental role

The Mathematics of Spin Spin seems to be a VECTOR: x y z

V z b y ^ x ^ y , V z = + a x ^ b y V z V z | |2 = 2 + 2 a b y ^ q 1 = What do we know about VECTORS? V z b y ^ x ^ y , Unit basis vectors: V z = + a x ^ b y Any vector: V z V z | |2 = 2 + 2 a b (Length)2: y ^ q 1 = 2 + 2 a b Unit vector: x ^ a x ^ Thus: a = cos q, b = sin q V z = cos q + sin q x ^ y And:

( ) _ _ _ B2 B2 B2 V z = cos q + sin q x ^ y V z = x ^ q = 0: V z = y Examples of: V z = x ^ q = 0: V z = y ^ q = 90: _ 1 B2 _ 1 B2 V z = y ^ x ^ q = 45: + 2 + 2 + a b = _ 1 B2 ( ) 2 = 1

x ^ y , = = x y z = a + b Application to Spin: Analogue of unit basis vectors: ? 1: “Quantization” of spin suggests: = = x y z = a + b ??

x y z = a + b 1 = 2 + 2 a b 2: Probability suggests a natural interpretation for: 1 = 2 + 2 a b

x y z = a + b 1 = 2 + 2 a b a2 b2 2: Probability suggests a natural interpretation for: 1 = 2 + 2 a b a2 b2

_ _ B2 B2 a2 = 0 b2 = 1 a2 = 1 b2 = 0 a2 = b2 = 1 _ 2 1 1 = + 100% 0% 50% 50% _ 1 B2 _ 1 B2 = +

_ B2 = + 1 = + 1 = + 1 q cos sin q _ 2 cos q sin q (?) (?) = + Summary: = + 1 = + _ 1 B2 = + 1 z q cos sin q _ 2 cos q sin q (?) (?) Question: y = + x

( ) ( ) _ B3 _ B3 q _ cos sin = + 2 q=60 1 = + 2 q=60 2 1 _ 2 Result: x y z q cos sin _ 2 Result: = + x y z q=60 B3 _ 2 1 E.g.: B3 _ 2 ( ) x y z q=60 = 75% 1 _ 2 ( ) = 25%

q _ cos sin = + 2 We have constructed a successful x y z q cos sin _ 2 We have constructed a successful “mathematical model” of spin! Does the mathematics predict anything interesting we have not thought of yet? The power of mathematics in the process of science

= 1 = + _ 1 B2 = 1 = + _ 1 B2 – = 1 – = 1 + q = 720:

= q = 0: = – q = 360: = + q = 720: The Mysterious Minus Sign: 1: Can be observed experimentally 2: Most important - sign in the universe 3: Has no “commonsense” interpretation

Spin is an excellent introduction to the “Quantum Information Age” So why might students care? (…assuming the previous reasons are not enough) Spin is an excellent introduction to the “Quantum Information Age”