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 &

• 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