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**The Physics of Spin Otto Stern Walther Gerlach 1922: Wrong theory**

right experiment Walther Gerlach

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**= N S Deflected Up Undeflected Beam of Atoms Oven Stern-Gerlach**

Detection Screen = Tiny Spinning Bar Magnet Silver Atom

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

weaker N S STONGER Deflected Up

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**= 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

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**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

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**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”:

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**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”

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**Summary of Experiments**

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

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**The Mathematics of Spin**

Spin seems to be a VECTOR: x y z

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**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:

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**( ) _ _ _ 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

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**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 ??

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

interpretation for: 1 = a b

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

interpretation for: 1 = a b a2 b2

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

50% 50% _ 1 B2 _ 1 B2 = +

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**_ 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

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**( ) ( ) _ 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%

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**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

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

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**= 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

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**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”

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