Transcript Document

The Physics of Spin
Otto Stern
1922: Wrong theory
 right experiment Walther Gerlach
Deflected
Up
Undeflected
Oven
Beam
of Atoms
Stern-Gerlach
Detection Screen
Silver
Atom
=
Tiny Spinning
Bar Magnet
=
N
S
Nonuniform
Magnetic
Field
N
weaker
STONGER
Deflected Up
S
=
N
S
Nonuniform
Magnetic
Field
N
weaker
STONGER
S
Undeflected
Experiment #1
50%
Random Spin
Directions
z
x
50%
ACTUALLY
EXPECT
SEE
TO (!!)
SEE
y
This result has no “classical”
or “commonsense” explanation
Experiment #2
100%
50%
1
Random
2
50%
0%
“Working
Hypothesis”
0%
It is consistent to hypothesize:
100%
Beam 1 atoms are all “spin up”:
Beam 2 atoms are all “spin down”:
Experiment #3
50%
Random
50%
HORIZONTAL
z
50%
VERTICAL
50%
HORIZONTAL
atom has 50/50 CHANCE
y
ofx 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:
z
y
x
What do we know about VECTORS?
z
V
y
b^
Unit basis vectors: ^
x,^
y
z ^
y
Any vector: V = ax + b ^
z2
2
(Length) : |V| = a2 + b2
z
V
^
y
q
^
x
^
ax
Unit vector: 1 = a2 + b 2
Thus: a = cos q, b = sin q
z
^
x + sin q
And: V = cos q^
y
z
^
Examples of: V = cos q^
x + sin q
y
q = 0:
z ^
V=x
q = 90:
z ^
V=y
_
_
z _
1 ^ _
1 ^
q = 45:
V=
y
x +
B2 B2
2
2
1
1
2
2
_
_
a +b =
+
= 1
B2
B2
_
_
() ()
Application to Spin:
Analogue of unit basis vectors: ^
x,^
y?
1: “Quantization” of spin suggests:
=
=
z
x
y
=
a
+ b
??
z
x
y
=
a
+ b
2: Probability suggests a natural
interpretation for: 1 = a2 + b 2
z
x
y
=
a
+ b
2: Probability suggests a natural
interpretation for: 1 = a2 + b 2
a2
b2
100%
0%
a2 = 1
0
100%
0%
0
b2 = 1
50%
a2
50%
=
__1
B2
+
__1
B2
1
_
=
2
1
_
2
b =
2
Summary:
=
1
+ 0
=
__1
+
__1
0
+ 1
=
B2
B2
z
q
Question:
x
_
q
_
q
y = cos
(?)q + sin
sin
(?) q
2
2
z
q
Result:
y
x
z
E.g.:
q=60
y
x
z
q=60
y
x
_
q
_
q
= cos
+ sin
2
2
_
B
3
_
=
2
+
_
1
2
_ 2
B_3 = 75%
( 2)
_
1
( 2)
2
= 25%
z
q
y
x
_
q
_
q
= cos
+ sin
2
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
_
_
+
B2
1
=
1
_
_
=–
B2
= –1
1
_
_
+
B2
q
= 720:
= +1
The Mysterious Minus Sign:
q
= 0:
=
q
= 360:
= –
q
= 720:
= +
1: Can be observed experimentally
2: Most important - sign in the universe
3: Has no “commonsense” interpretation
So why might students care?
(…assuming the previous reasons are not enough)
Spin is an excellent introduction
to the “Quantum Information Age”