Why is Quantum Mechanics different

Why is Quantum Mechanics different?

Why is Quantum Mechanics different?

Different Abstractions

States and Measurements

An Experiment with Spins and Qubits

Spin: Spin Angular Momentum

Property in addition to Spatial location

An extra Degree of Freedom for electrons

Qubit: Two State 'Quantum' System

What does an Electron's Spin look like?

x

y

z

electron

S


x

y

z

?

S

Apparatus

(This Side Up)

Spin

Before Measurement


x

y

z

+1/-1

S

Apparatus

(This Side Up)

+1

After Measurement

Sz=+1 or -1

or

-1

S


x

y

z

+1

S

Apparatus

(This Side Up)

Spin

After MeasurementConsider the case Sz=+1

x

y

z

+1

Apparatus

(This Side Up)

Spin

After MeasurementConsider the case Sz=+1We measure Sz Once Again

?

x

y

z

+1

Apparatus

(This Side Up)

Spin

After MeasurementConsider the case Sz=+1We measure Sz Once Again

+1

Spin

x

y

z

+1

Apparatus

(This Side Up)

Spin

After MeasurementConsider the case Sz=+1We measure Sz One more time

?

x

y

z

+1

Apparatus

(This Side Up)

Spin

After MeasurementConsider the case Sz=+1We measure Sz One more time

+1

Spin

x

y

z

+1

Apparatus

(This Side Up)

Spin


Next we Measure Spin with Apparatus topside down

?

x

y

z

+1

Apparatus

(This Side Up)

Spin


-1

Spin

Next we Measure Spin with Apparatus topside down

x

y

z

+1

Apparatus

Spin


We Measure Spin with Apparatus Sideways along y

?

x

y

z

+1

Apparatus

Spin


0 Spin 0

We Measure Spin with Apparatus Sideways along y

x

y

z

+1

Apparatus

Spin


0 Spin 0

We Measure Spin with Apparatus Sideways along yX

x

y

z

+1

Apparatus

Spin


+1 Spin

We Measure Spin with Apparatus Sideways-1 Spin

OR

Sy=+1/-1

x

y

z

+1

Apparatus

Spin+1 Spin

We Measure Spin with Apparatus Sideways

-1 Spin

OR

Sy=+1/-1 with Equal Probability

(50%)

(50%)

x

y

z

?

Apparatus

Spin

+1

Spin

We Rotate the Apparatus by an Arbitrary Angle

-1

Spin

+/-1

m

mn

x

y

z

+1

Spin

We Rotate the Apparatus by an Arbitrary Angle

-1

Spin

mn

S is extremely Stubborn:Returns only +1/-1 after measurement

<S>= n.m

However the Average Spin <S> varies with angle between n and m

x

y

z

?

Apparatus

Spin

Consider the case Sy = +1 after 1st MeasurementAnd Sz = +1 after 2nd Measurement

+1Spin

We Measure Spin : Sideways along y first and then Upwards along Z

+1

Spin Sz=+1Sy= ?

x

y

z

?

Apparatus

Spin

+1Spin

We Measure Spin : Sideways along y first and then Upwards along ZWe measure Sy all over again.

Should Sy differ from initial measurement(+1)?

+1

Spin

+1 Spin

?

x

y

z

?

Apparatus

Spin

+1Spin

We Measure Spin : Sideways along y first and then Upwards along Z, andSideways along y once again

+1

Spin

+1 Spin

+1 -1

Experiments are never Gentle, MeasurementsDistort systems.

Classical Experiments can be 'Arbitrarily' gentle.

In Quantum Mechanics, any interaction strong enough toMeasure some aspect of the system is necessarilystrong enough to disrupt some other aspect of the same system.

Sx,Sy and Sz can never be measured simultaneously.Measuring one component of the Spin disrupts the othertwo components.

Spin of an Electron is a Quantum phenomenon.There exists no Classical Analogue for an Electron's Spin.

[A,B]=AB – BA : Commutator of A and B

For Sx, Sy, Sz, [Si,Sj] is non zero : Si and Sj donot 'Commute'.Si : Denotes electron's Spin measurement along the 'i' direction.

More Generally, A and B :Denote some Property 'A' and 'B'Of the Quantum System.

[A,B] = non zero : Properties A and B for the System cannever be measured simultaneously.

Non Commutation Relations are the Hallmark ofQuantum Systems.

x

y

z

?

Apparatus

Spin

+1Spin

We Measure Spin : Sideways along y first and then Upwards along Z, andSideways along y once again

If X 'denotes' my system, SzSyX equivalentlydescribes the procedure above

+1

Spin

Sz

Sy

Sy

SzSyX


Heisenberg's Uncertainity relation:

( ΔA)ψ( ΔB)ψ ≥ |<ψ|[A,B]|ψ>|/2


Heisenberg's Position-Momentum Uncertainity:

( Δx)ψ( Δp)ψ ≥ |<ψ|[x,p]|ψ>| /2


Heisenberg's Uncertainity relation:

( Δx)ψ( Δp)ψ ≥ h/2Position-Momentum Uncertainity relation

Probing Wave-Particle Duality

An Experiment with Bullets

Gun

Slits Detector

Bullets are discrete particles

y y

1

2



Gun

Slits Detector

P11

y

.

.

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Gun

Slits Detector

P22

y

.

.

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Gun

Slits Detector

P12P1

P2

y y



Gun

Slits Detector

P12P1

P2

y y

P12 = P1 + P2

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Probing Wave-Particle DualityAn Experiment with Water Waves

WaveSource

Slits Detector

I12

I1

I2

y y

I12 = I1 + I2/I12 ~ P12

?

Water Waves flow in Continuum


WaveSource

Slits Detector

I12

I1

I2

y y

I12 = I1 + I2/I12 ~ P12


WaveSource

Slits Detector

I12

I1

I2

y y

I12 = I1 + I2/I12 ~ P12

I12 = I1 + I2 + 2√I1I2 cosδ


An Experiment with Electrons

Electron Source

Slits Detector

y y

Electrons are discrete particles, like Bullets



ElectronSource

Slits Detector

P11

y

.

.

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ElectronSource

Slits Detector

P22

y

.

.

..

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..



Electron Source

Slits Detector

P12P1

P2

y y

P12 = P1 + P2

..

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Electron Source

Slits Detector

P12P1

P2

y y

P12 = P1 + P2

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Electron Source

Slits Detector

P12P1

P2

y y

P12 = P1 + P2

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Electron Source

Slits Detector

P12P1

P2

y y

P12 = P1 + P2

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\Electron Probabilities for 1 and 2 donot add up, unlike Bullets.

Electrons interfere like Waves.

However, unlike Waves(continous), Electrons are discrete and 'Lumpy'.


A different Electron Experiment

Electron Source

Slits

Detector

P12P1

P2

y y

We Repeat the Electron Expt. With a Difference:



Electron Source

Slits

Detector

P12P1

P2

y y


A Laser Scintillator



Electron Source

Slits

Detector

P12P1

P2

y y

P12 = P1 + P2

LaserScintillator





Electron Source

Slits

Detector

P12P1

P2

y y

LaserScintillator



?



Electron Source

Slits

Detector

P12P1

P2

y y

P12 = P1 + P2

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LaserScintillator





Electron Source

Slits

Detector

P12P1

P2

y y

P12 = P1 + P2

Measurement 'Destroys' the Interference Pattern for Electrons.

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LaserScintillator



Electron Source

Slits

Detector

P12P1

P2

y y

P12 = P1 + P2

Measurement 'Destroys' the Interference Pattern for Electrons.

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LaserScintillator

Quantum Mechanics : Quantum Superposition

Classical Superposition doesnot work here.



Electron Source

Slits Detector

P12P1

P2

y y

P12 = P1 + P2

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\Electron Probabilities for 1 and 2 donot add up, unlike Bullets.

Instead Probability Amplitudes add up: This is Quantum Superposition



Electron Source

Slits Detector

P12P1

P2

y y

P12 = P1 + P2

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\And Measurements on a State, and the State of a System are not same.


Electrons vs Bullets

P12 electrons

y y

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Electrons vs Bullets

P12 electrons

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y y

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Why is Quantum Mechanics different