Quantum Chicken Rice
29
TWO-LEVEL SYSTEMS
description
macroni chicken
Transcript of Quantum Chicken Rice
TWO-LEVEL SYSTEMS
WAVE FUNCTIONS ARE NOT ENOUGH
Stern-Gerlach experiment Electron has an intrinsic angular momentum called spin Description of spin is outside domain of wave functions
STERN-GERLACH SETUP
ORBITAL MAGNETIC DIPOLE MOMENT
Magnetic dipole moment and orbital angular momentum
1
I =e
T=
ev
2⇡r(1)
this current produces a magnetic field that from far away looks just like the field from a bar magnet with a magnetic dipole moment
1
I =e
T=
ev
2⇡r(1)
µ = IA = I⇡r2 (2)
µ =evr
2=
e
2mL (3)
~µ = � e
2m~L (4)
1
I =e
T=
ev
2⇡r(1)
µ = IA = I⇡r2 (2)
µ =evr
2=
e
2mL (3)
~µ = � e
2m~L (4)
1
I =e
T=
ev
2⇡r(1)
µ = IA = I⇡r2 (2)
µ =evr
2=
e
2mL (3)
~µ = � e
2m~L (4)
angular momentum
INTERACTION WITH EXTERNAL MAGNETIC FIELD
The potential energy of interaction (energy of orientation)
1
I =e
T
=ev
2⇡r(1)
µ = IA = I⇡r
2 (2)
µ =evr
2=
e
2mL (3)
~µ = � e
2m~
L (4)
V = �~µ · ~B (5)
~
F = �~rV (6)
F
x
= ~µ · @~
B
@x
(7)
F
y
= ~µ · @~
B
@y
(8)
F
z
= ~µ · @~
B
@z
(9)
~⌧ = ~µ⇥ ~
B (10)
L = n~ (11)
µ =e~2m
n (12)
Translational force acting on the bar magnet:
1
I =e
T
=ev
2⇡r(1)
µ = IA = I⇡r
2 (2)
µ =evr
2=
e
2mL (3)
~µ = � e
2m~
L (4)
V = �~µ · ~B (5)
~
F = �~rV (6)
F
x
= ~µ · @~
B
@x
(7)
F
y
= ~µ · @~
B
@y
(8)
F
z
= ~µ · @~
B
@z
(9)
~⌧ = ~µ⇥ ~
B (10)
L = n~ (11)
µ =e~2m
n (12)
1
I =e
T
=ev
2⇡r(1)
µ = IA = I⇡r
2 (2)
µ =evr
2=
e
2mL (3)
~µ = � e
2m~
L (4)
V = �~µ · ~B (5)
~
F = �~rV (6)
F
x
= ~µ · @~
B
@x
(7)
F
y
= ~µ · @~
B
@y
(8)
F
z
= ~µ · @~
B
@z
(9)
~⌧ = ~µ⇥ ~
B (10)
L = n~ (11)
µ =e~2m
n (12)
1
I =e
T
=ev
2⇡r(1)
µ = IA = I⇡r
2 (2)
µ =evr
2=
e
2mL (3)
~µ = � e
2m~
L (4)
V = �~µ · ~B (5)
~
F = �~rV (6)
F
x
= ~µ · @~
B
@x
(7)
F
y
= ~µ · @~
B
@y
(8)
F
z
= ~µ · @~
B
@z
(9)
~⌧ = ~µ⇥ ~
B (10)
L = n~ (11)
µ =e~2m
n (12)
1
I =e
T
=ev
2⇡r(1)
µ = IA = I⇡r
2 (2)
µ =evr
2=
e
2mL (3)
~µ = � e
2m~
L (4)
V = �~µ · ~B (5)
~
F = �~rV (6)
F
x
= ~µ · @~
B
@x
(7)
F
y
= ~µ · @~
B
@y
(8)
F
z
= ~µ · @~
B
@z
(9)
~⌧ = ~µ⇥ ~
B (10)
L = n~ (11)
µ =e~2m
n (12)
CLASSICAL PREDICTION
Thermal atoms have randomly oriented magnetic moments
since the field has a gradient along the vertical axis, it is expected that atoms will get deflected along this direction and different atoms by different amount so that finally we should see on the screen something similar to the right panel
EXPERIMENTAL DATA
Magnetic moment is quantized In conclusion so is angular momentum
And this result is the same for various orientations of the magnet.
POSTCARD TO BOHR
HAS IT REALLY BEEN CONFIRMED...?
Back to Bohr model
angular momentum is quantized l = 1,2,3,...
1
I =e
T
=ev
2⇡r(1)
µ = IA = I⇡r
2 (2)
µ =evr
2=
e
2mL (3)
~µ = � e
2m~
L (4)
V = �~µ · ~B (5)
~
F = �~rV (6)
F
x
= ~µ · @~
B
@x
(7)
F
y
= ~µ · @~
B
@y
(8)
F
z
= ~µ · @~
B
@z
(9)
~⌧ = ~µ⇥ ~
B (10)
L = l~ (11)
L
z
= m~ (12)
µ =e~2m
n (13)
suggest that the components are also quantized ml = -l, -l+1,...,0,1,..., l
1
I =e
T
=ev
2⇡r(1)
µ = IA = I⇡r
2 (2)
µ =evr
2=
e
2mL (3)
~µ = � e
2m~
L (4)
V = �~µ · ~B (5)
~
F = �~rV (6)
F
x
= ~µ · @~
B
@x
(7)
F
y
= ~µ · @~
B
@y
(8)
F
z
= ~µ · @~
B
@z
(9)
~⌧ = ~µ⇥ ~
B (10)
L = l~ (11)
L
z
= m
l
~ (12)
µ
z
= � e~2m
m
l
= �µ
B
m
l
(13)
1
I =e
T
=ev
2⇡r(1)
µ = IA = I⇡r
2 (2)
µ =evr
2=
e
2mL (3)
~µ = � e
2m~
L (4)
V = �~µ · ~B (5)
~
F = �~rV (6)
F
x
= ~µ · @~
B
@x
(7)
F
y
= ~µ · @~
B
@y
(8)
F
z
= ~µ · @~
B
@z
(9)
~⌧ = ~µ⇥ ~
B (10)
L = l~ (11)
L
z
= m
l
~ (12)
µ
z
= � e~2m
m
l
= �µ
B
m
l
(13)corresponding quantization of magnetic moment
Bohr magneton
1
I =e
T
=ev
2⇡r(1)
µ = IA = I⇡r
2 (2)
µ =evr
2=
e
2mL (3)
~µ = � e
2m~
L (4)
V = �~µ · ~B (5)
~
F = �~rV (6)
F
x
= ~µ · @~
B
@x
(7)
F
y
= ~µ · @~
B
@y
(8)
F
z
= ~µ · @~
B
@z
(9)
~⌧ = ~µ⇥ ~
B (10)
L = l~ (11)
L
z
= m
l
~ (12)
µ
z
= � e~2m
m
l
= �µ
B
m
l
(13)
µ
B
= 9.27⇥ 10�24 J/T (14)
Conclusion: there always is an odd number of possible values for a component of orbital magnetic dipole moment.
SPIN
There is a new source of angular momentum Stern-Gerlach result is not due to orbital angular momentum Let us postulate existence of another angular momentum
2
µ
z
= � e~2m
m
l
= �µ
B
m
l
(13)
µ
B
= 9.27⇥ 10�24 J/T (14)
~µ
s
= � e
2m~
S (15)
S
z
= m
s
~ = ±1
2~ (16)
z component of spin (intrinsic angular momentum)
from the experiment: only two opposite sign components equal to 1/2 hbar as we assume that the difference in ms is 1, just as for orbital angular momentum
ELECTRON IS SPIN 1/2 PARTICLE
Spin quantum number Assuming that allowed spin components are from -s to s, s = 1/2.
Spin gyromagnetic ratio Relation between spin magnetic moment and spin angular momentum
2
µ
z
= � e~2m
m
l
= �µ
B
m
l
(13)
µ
B
= 9.27⇥ 10�24 J/T (14)
~µ
s
= � e
2m~
S (15)
S
z
= m
s
~ = ±1
2~ (16)
s =1
2(17)
µ
s
= �g
s
µ
B
m
s
(18)
spin gyromagnetic ratio very close to 2, established experimentally
WHY IS THIS CALLED SPIN?
The Abraham-Lorentz model Electron is a charged rotating (spinning) sphere
Problems of this classical picture (including relativistic effects) 1. Spin velocity essentially equal to the speed of light 2. Unstable: radiates energy away, and fast...
SIMPLEST QUANTUM SYSTEM
Spin-1/2 has only two possible measurement outcomes
Other examples of two level systems Polarization of single photon A particle propagating along only two accessible paths
POLARIZATION
Describes the way electric vector behaves There are only two distinguishable polarization states
This is linearly polarized light as the electric vector oscillates in a plane. To be precise this is vertically polarized light. Fully distinguishable from horizontally polarized light.
POLARIZATION
This is again linearly polarized light but in another plane. Polarization distinguishable to this one is along orthogonal axis. Note that we can decompose this polarization in terms of horizontal and vertical.
There are many pairs of distinguishable polarizations
POLARIZATION
Circular polarization The phase between H and V is ±pi/2
Right-handed circular polarization. Left-handed circular polarization.
POLARIZER
Device that passes light of a specific polarization
LCD NUMERICAL DISPLAY
The black electrodes are transparent to light and if they are not charged the liquid crystals between them rotate polarization by 90 degrees, such that it goes through the second polarizer.
THE MALUS LAW
Governs intensity of light passing through two polarizers
IT IS HARD TO MAKE PHOTONS INTERACT
BACK TO MALUS
Every single photon either goes through or not This happens with certain probability given by... ... the general Born rule
2
µ
z
= � e~2m
m
l
= �µ
B
m
l
(13)
µ
B
= 9.27⇥ 10
�24J/T (14)
~µ
s
= � e
2m
~
S (15)
S
z
= m
s
~ = ±1
2
~ (16)
s =
1
2
(17)
µ
s
= �g
s
µ
B
m
s
(18)
~v = cos ✓ ~p+ sin ✓ ~p? (19)
COLLAPSE OF THE WAVE FUNCTION
Polarizer is a measuring device It has two outcomes: passes the photon or absorbs it
Collapse Whatever polarization of incoming photon if it passes through... ... it has to be consistent with the polarizer direction
ANOTHER EXAMPLE OF TWO-LEVEL SYSTEM
Photon that can only propagate horizontally or vertically Because we force it with mirrors and glasses (beam-splitters)
WINDOW GLASS IS A BEAM SPLITTER
You can see through and you can see your reflection Each single photon is superposed to go through and reflect
CLAUSER EXPERIMENT
Prediction of classical electromagnetism Both detectors may click even for small light intensity
light source
attenuator
50-50 beam splitter
QUANTUM COIN TOSSING
For weak light only one detector clicks In agreement with photon picture
Which detector will click in the next run? This cannot be predicted
What can be predicted? How many times a detector will click in many runs
QUANTUM INTERFERENCE
Interference of single quantum Photons, electrons,... interfere with themselves
When does it happen? When there are indistinguishable ways to reach detector
QUANTUM SEEING IN THE DARK
SUMMARY
Wave functions do not provide complete info of a system
It is convenient to describe 2-level system as 2d vectors
