# ch39 young freedman - Waves As we have seen, light has a duality of being a wave and a particle. By a symmetry argument, de Broglie in 1924 proposed that all form of matter should

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Heisenbergs Uncertainty Principle

Probability in finding where they land requires a wave description

Photons going through the slit as particles

THEN the spreading-out of the photons implies that the photons reaching the screen must have a small uncertainty in the vertical ( ) direction of its momentum as it exits the slit.y

yp

We will try to estimate the uncertainty for these photons by

Heisenbergs Uncertainty Principle

yp

yp

The central max dominates the diffraction pattern and a majority of the photons will land in the central max.

Considering the diffraction pattern, the width of the central max is given by the location of its 1st order minimum ,

Heisenbergs Uncertainty Principle

1sin a

1

Assuming that so that is small so that we have 1a 1 1sin a

yp

yp

- And, from the diagram,

Heisenbergs Uncertainty Principle

- For the photons reaching , they must have a range of y-dir momentum:

,y yp p

1y xp p

So, we can estimate the uncertainty needs to be as large as 1xp yp

1

1

1 1tany

x

pp

for small

Heisenbergs Uncertainty Principle

With Then, putting in , we have 1y x xp p p a 1 a

The incoming photons moving along the x-dir with momentum in the x-dir

will have a wavelength given by . Substituting this into the

inequality, we have

xp

y xp p x

hp

haa

xh p

1y xp p

yp

Heisenbergs Uncertainty Principle

For photons going through the slit with width a, one can also say that the y-

position of the photons has an uncertain of

yy p h

y a

Putting everything together, we arrive at a constraint relating the conjugate pair

of dynamic variables such that, , yy p (where h is the Plancks constant)

yp

Heisenbergs Uncertainty Principle

Our discussion is not specific to the direction. In general, we will have

similar uncertainty relations for all three spatial directions, i.e.,

2yy p

We have use heuristic argument in getting to previous inequality. A more

precise inequality can be derivate from full QM calculations and it should read

where / 2h

, ,2 2 2x y z

x p y p z p

y

This is called the Heisenbergs Uncertainty Principle.

But, there are no restrictions for unconjugated variables: , .yx p or x y etc

Uncertainty Relation for E & t

Intuitively, the Heisenbergs Uncertainty Principle enforces an inverseproportional relation on the two conjugate pairs of dynamics variables:

,xp x ,E t and

By decreasing the uncertainty in one of the variables (x or t), its corresponding conjugate variable must increases accordingly ! orxp E

2xp x

2E t

And, additionally, there is an uncertainty relation for and :E t

similar to

Heisenberg Uncertainty Principle againThe Heisenberg Uncertainty Principle is not a statement on the accuracy of the experimental instruments. It is a statement on the fundamental limitations in making measurements !

To understand this, there are two important factors:

1. The wave-particle duality2. The unavoidable interactions between the observer and the object

being observed

Uncertainty Principle: Conceptual ExampleSuppose we want to measure the position of a particle (originally at rest) by a laser light.

The measurement is accomplished by the scattering of photons off this particle (shining a light) :

photon

particle

scattered phone

Uncertainty Principle: Conceptual ExampleBut, photons carry momentum and the particle after scattering will recoil !

particlescattered phone

recoil

While the scattered photon gives us a precise position of the particle, the photon will also inevitably impart momentum onto the particle.

Uncertainty Principle: Conceptual ExampleIf the measurement is made by a photon with wavelength , then at best,

we expect our position measurement to be accurate up to an uncertainty,

x

And, the photon will transfer parts of its momentum to the particle along the measurement direction x. The amount transfer is given by,

xhp

Combining these two expressions gives: xx p h

Note: By improving x using a smaller does not help because px ~ h/gets larger in inverse proportion to !.

Waves and UncertaintyConsider a plane wave with polarization in the y-dir and propagating in the x-dir:

, sinyE x t A kx t

at a fixed time: 0 0t t ,0 siny kxE x A

at a fixed location: 0 0x x 0, sinyE t A t

x

yE

t

yE

wave with a definite wavelength :2 k

wave with a definite frequency :2f

f

Waves and UncertaintyRecall that is the wave number and is the angular frequency of the wave and we can express the x-momentum and energy of a photon in terms of

22x

hp kh

k

andk

22

hf fE h

and

Substituting these relations into our equation for , ,yE x t

, sinyE x t A kx t

This wave function for a (plane wave) photon expressed in this way will have definite values of x-momentum and energy , i.e.,xp E

0xp 0E and

There is no uncertainty in the values for and for a plane wave.xp E

, siny xp EE x t A x t we have,

x

yE

Waves and UncertaintyOne might then ask if , can be still be satisfied?

YES! In fact, the Heisenbergs Uncertainty Principle implies that:

so that the uncertainty in the position for a plane wave must be infinite !

0xp

x

Looking back at our plane wave with a definite wavelength (x-momentum ), it indeed extends over the entire range of x and one cannot localize where the photon might be.

2xp x

xp

t

yE

Waves and UncertaintyCorrespondingly, one can argue that a similar uncertainty relation applies to the uncertainties in energy and time, i.e.,

And, in particular, for a plane wave, we have,

For a plane wave with a definite frequency (energy E), it will extend over all time t and one cannot localize the photon in time.

2E t

0E t and

(Heisenbergs Uncertainty Principle for energy and time)

Waves and UncertaintyObviously, a plane wave with infinite extents in space & time is an idealization.

In practice, wave function of a photon will have limited extents in space & time (i.e., we know roughly where it is located in space and time),

x t and

Then, correspondingly, according to the Heisenbergs Uncertainty Principle, the uncertainties for the x-momentum and energy for a realistic photon cannot be zero,

0xp 0E and

such that and are satisfied !2x

p x 2

E t

Waves and Uncertainty

Intuitively, the Heisenbergs Uncertainty Principle enforces an inverseproportional relation on the two conjugate pairs of dynamics variables:

,xp x ,E tand

By decreasing the uncertainty in one of the variables (x or t), its corresponding conjugate variable must increases accordingly ! orxp E

2xp x

2E t

Waves and UncertaintyTo visualize a localized wave function, one can consider a superposition of multiple plane waves

1 1 1 2 2 2, sin siny x xE x t A p x E t A p x E t

For simplicity, let consider the sum of two of the same plane waves that we have been considering BUT with slightly different x-momentum and E.

Looking at the wave at a fixed time and0 0t t

2 1A A we have

(black) (red)(green)

Waves and Uncertainty

The resultant electric field are localized (beat pattern). With superposition of waves with many wavelengths/frequencies (x-momentum/energy), the localization can be stronger. The result is called a wave packet.

NOTE: This wave packet is localized in space BUT it contains many and as required by the HUP.

x xp 0xp

PHYS 262George Mason University

Prof. Paul So

Chapter 39: Particles Behaving as Waves Matter Waves Atomic Line-Spectra

and Energy Levels Bohrs Model of H-

atom The Laser Continuous Spectra &

Blackbody Radiation

Matter WavesAs we have seen, light has a duality of being a wave and a particle.

By a symmetry argument, de Broglie in 1924 proposed that all form of matter should process this duality also.

Recall for photons, we have:

For a particle with momentum p = mv (or mv) and total energy E, de Broglie proposed:

h h hp mv mv

Efh

(de Broglie wavelength)

hp

Electron DiffractionClinton Davisson and Lester Germer (1927)

sin 50od

Nickel crystal acts as a diffraction grading.

Constructive interference is expected at:

sin , 1, 2,d m m

2 / 2

/ / 2a

a

eV p m

h p h meV

54aV V peak

Va

Diffraction of e off Aluminum FoilIn 1928, G.P. Thomson (son of J.J. Thomson) performed another demonstrative experiment in showing electrons can act as a wave and diffract from a polycrystalline aluminum foil.

Debye and Sherrer: X-Ray diffraction experiment done a few years earlier using a similar set

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