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

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

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

4. = 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

5. 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

6. 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”:

7. 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”

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

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

10. 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 = a b
(Length)2: y ^ q
1 = a b
Unit vector: x ^ a x ^
Thus:
a = cos q, b = sin q V z
= cos q + sin q x ^ y
And:

11. ( ) _ _ _ 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: + + a b = _ 1 B2
( ) 2
= 1

12. 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 ??

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

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

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

16. _ 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

17. ( ) ( ) _ 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%

18. q _ cos sin = + 2 We have constructed a successfulx 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

19. = 1 = + _ 1 B2 = 1 = + _ 1 B2 – = 1 – = 1 +
q = 720:

20. = 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

21. 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”