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Published byBret Trapp Modified about 1 year ago

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

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

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

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

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

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

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

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Summary of Experiments 100% 50% 0% 100% 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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What do we know about VECTORS? y ^ x ^ ax ^ b y ^ V z Unit basis vectors: x ^ y ^, Any vector: V z =+ ax ^ b y ^ V z | | 2 = 2 + 2 ab (Length) 2 : Unit vector: 1 =1 = 2 + 2 ab Thus: a = cos q, b = sin q V z q And: V z = cos q + sin q x ^ y ^

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V z = y ^ Examples of: V z = cos q + sin q x ^ y ^ _ 1 B2B2 _ _ 1 B2B2 _ V z = x ^ q = 0:q = 90: V z = y ^ q = 45: x ^ + 2 + 2 + ab = _ 1 B2B2 _ ( ) 2 _ 1 B2B2 _ 2 = 1

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

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

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x y z = a+b 2: Probability suggests a natural interpretation for: 1 =1 = 2 + 2 ab b2b2 a2a2

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100% 50% 0% 100% a 2 = 1 b 2 = 0 a 2 = 0 b 2 = 1 = + _ 1 B2B2 _ _ 1 B2B2 _ a2 =b2 =a2 =b2 = 1 _ 2 1 _ 2

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cos q sin q Summary: = + _ 1 B2B2 _ _ 1 B2B2 _ = + 10 Question: = + 01 = + x y z q (?) (?) cos sin q_ 2 q_ 2

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Result: = + x y z q cos sin q_ 2 q_ 2 E.g.: = + x y z q=60 B3B3 _ 2 _ 1 _ 2 = 25% = 75% x y z q=60 B3B3 _ 2 _ ( ) 2 1 _ 2 2

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We have constructed a successful “mathematical model” of spin! = + x y z q cos sin q_ 2 q_ 2 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 B2B2 _ _ 1 B2B2 _ = 1 = 1 =+ _ 1 B2B2 _ _ 1 B2B2 _ – = 1– = 1+q = 720:

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

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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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