Science One Physics Lecture 7

The Heisenberg Uncertainty Relation

Quantum measurement

determinism vs.probability

particle - waveduality

Heisenberguncertainty relation

bound states

entanglement

SchrodingerEquation

:

superpositionprinciple

Outline

• The double-slit experiment, revisited

• Measurement in quantum mechanics

• ~: The birth of quantum mechanics

• The Heisenberg uncertainty relation

• The Schrodinger equation

• Entanglement and quantum cryptography

Where does quantum mechanics begin?

water waves (always)

sourcelight (Thomas Young, 1801)

neutrons (A. Zeilinger et al. 1988) observation

What is the observed interference pattern an image of?

Where does quantum mechanics begin?

What is the observed interference pattern an image of?

observation|A(x)| 2,

|ψ(x)|2

x

water waves: |A(x)|2 ∼ ε(x) (energy density),

bright light: |A(x)|2 ∼ ε(x) (energy density),

single photons: |ψ(x)|2 = p(x) (probability density),

neutrons: |ψ(x)|2 = p(x) (probability density).

Where does quantum mechanics begin?

The normalization condition:

Classical waves: water, earthquakes, guitar strings, bright light:∫|A(x)|2dx ∼ Energy, (unbounded)

Quantum waves: individual photons, neutrons, buckyballs, ...:∫|ψ(x)|2dx = 1

This has implications for measurement: If you measure a particle

to be “here”, then it can’t be “there”, too.

Measurement in quantum mechanics (I)

• Classically, a particle is at position x, and by measurementwe can find out exactly what x is.

• Quantum mechanically, a particle is described by a wavefunction ψ(x).

How do we assign a position to it?

The position is truly random. A measurement finds the par-ticle in the spot x0 with probability density |ψ(x0)|2.

Measurement in quantum mechanics (I)

• Classically, a particle is at position x, and by measurementwe can find out exactly what x is.

• Quantum mechanically, a particle is described by a wavefunction ψ(x).

How do we assign a position to it?

The position is truly random. A measurement finds the par-ticle in the spot x0 with probability density |ψ(x0)|2.

What happens if we measure again?

• Classically, if the particle is found at position x in the first

measurement, it is found at x in the second measurement,

with certainty.

• Quantum mechanically, ... it is exactly the same.

It this wasn’t so, then what would be the meaning of measure-

ment?

What happens if we measure again?

• Classically, if the particle is found at position x in the first

measurement, it is found at x in the second measurement,

with certainty.

• Quantum mechanically, ... it is exactly the same.

It this wasn’t so, then what would be the meaning of measure-

ment?

Implication

Quantum measurement changes the measured state.

• Classically, if the particle is found at position x in the first

measurement, it is found at x in the second measurement,

with certainty.

• Quantum mechanically, ... it is exactly the same.

Collapse of the wave function

measuremen

t

In quantum mechanics, measurement is a dynamical process in

which the wave function undergoes a drastic change.

The founders – Max Planck

The founders – Max Planck

Max Planck in 1901

Planck’s problem

Rayleigh-Jeans law(classical explanation)

Ultraviolet catastrophe

Inte

nsity

per

wav

elen

gth

wavelengthλ, μm

Planck’s quantum hypothesis (1900)

Light caries energy in indivisible packets, or quanta.

For light of frequency f , each quantum has an energy of

E = hf

Planck’s law of black body radiation (1900)

Using the quantum hypothesis plus Boltzmann’s factor describing

the suppression of high-energy states, prob(ν) ∼ e−Eν/kT :

I(λ) =2hc2

λ5

1

ehcλkT − 1

Resolution of the UV catastrophe

Rayleigh-Jeans law(classical explanation)

Planck’s law(new explanation)

Ultraviolet catastrophe

Inte

nsity

per

wav

elen

gth

wavelengthλ, μm

h is small

The numerical value of Planck’s constant is

h = 6.626070040× 10−34 Js

What does “small” mean?

• 1 eV: Kinetic energy of an electron acquired in a voltage

drop of 1V.

• 0.035 eV: Average kinetic energy of an atom at room tem-

perature (3/2 kBT )

• 2.4 eV: Energy of a single photon from the middle of the

visible spectrum (600 THz).

⇒ The energy of a single photon is “typical” for microscopic

phenomena.

Quanta are real: the photoelectric effect

f

Einstein’s explanation (1905): Quanta of light transfer their en-ergy E = hf to individual electrons, kicking them out.

Confirmation of Planck’s quantum hypothesis

DeBroglie: Matter can behave as a wave (1924)

p = hλ

Particle-wave duality

Davisson-Germer experiment (1927)

Peaks of electron intensity at distinct angles

Same as with photon scattering!

Nickel

Experimental confirmation of the wave nature of particles

Where does quantum mechanics begin?

water waves (always)

sourcelight (Thomas Young, 1801)

neutrons (A. Zeilinger et al. 1988) observation

The De Broglie relation

• Assigns a momentum p to a photon of a given wavelength λ

• Assigns a wavelength λ to a neutron of momentum p

• Doesn’t apply to water waves

Quantum measurement (II)

What else can we measure besides position?

What is the wavelength of this wave function?

(what does this even mean?)

Quantum measurement (II)

Analogy with a guitar string:

The amplitude A(x) of a guitar string of length L, at time t0,can be expressed as

A(x) =∑nB(n) sin

(πn

x

L

).

The function A(x) and the amplitudes {B(n), n ∈ N} contain the same infor-

mation, but the latter tell us more directly how the string sounds.

Likewise we may expand the wave function:

ψ(x) =∫ψ(λ) sin

(2πx

λ

)dλ.

Therein, |ψ(λ)|2 is the probability density for finding the value λ

in a measurement of the wavelength.

Quantum measurement (II)

We may expand the wave function ψ(x) into sinusoidal waves,

ψ(x) =∫ψ(λ) sin

(2πx

λ

)dλ.

Therein, |ψ(λ)|2 is the probability density for finding the value λ

in a measurement of the wavelength.

By the De Broglie relation, p = h/λ, a measurement of wave-

length λ amounts to a measurement of momentum p.

Thus, besides position, we can also measure momentum.

Quantum measurement (II)

Can we measure position and momentum simultaneously arbi-

trarily sharply?

Classically, Yes.

Quantum mechanically, No.

If position is measured first, then this measurement will change

the momentum distribution in the wave function, which affects

a subsequent momentum measurement; and vice versa.

The Heisenberg uncertainty relation

The impossibility to measure position and momentum simultane-

ously sharply is captured in the Heisenberg Uncertainty Relation,

∆x∆p ≥~2

Therein: ∆x (∆p): uncertainty in position (momentum); ~ = h/2π.