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Lecture 7. Heisenberg Uncertainty Principle

- Outline
- Wave Packets
- Uncertainty position ? momentum (1927)
- Uncertainty energy ? time

The uncertainty principle saves us from

contradiction between the complimentary wave-like

and particle-like properties of quantons.

Wave Packets

Plane monochromatic wave describes the particles

state with a well-defined momentum p hk

According to the statistical interpretation of

the wave function, the probability to find the

particle within ?x is proportional to

- thus, the probability is constant

(x-independent), the particle is unlocalized!

snapshot of the distribution A(x,t) at t t0

To localize the particle, we need to form a wave

packet (group), by combining plane waves with

different ks.

Because of the dispersion of dB waves, the packet

will spread with time. Again, it doesnt mean

that the particle itself spreads out with time,

rather, the probability of finding the particle

away from a moving center of the distribution

A(x,t) grows with time.

Localization of the Wave Packet

To localize a quanton, we need to prepare a

superposition of plane waves with some non-zero

range of k values,

the result of addition

Now the momentum is now well-defined it is

somewhere within the range

Thus, for a wave packet

Uncertainty position ? momentum

More accurate definition of ?x for an arbitrary

distribution A(x), ?x is the standard deviation

for this distribution (the root-mean-square

deviation of the values from their mean)

A quantons position and momentum cannot be

simultaneously well-defined their uncertainties

?x and ?px are inter-related

Uncertainty of the momentum

Uncertainty of the position

- The U.P. couples only incompatible

observables (every pair of observables whose

operators do not commute)

- The indeterminacy reflects the limits beyond

which we can no longer use the intuitive

classical notions of the particles trajectory.

- In quantum mechanics, the division of the total

energy of a particle into its kinetic (p) and

potential (x) parts is also blurred.

The complimentary wave-like and particle-like

properties of quantons can be reconciled only

within limits imposed by the uncertainty

principle.

Electron Diffraction

Lets consider a plane dB wave that describes an

electron with a well-defined momentum p (along

y). To determine the electron position along the

x-axis, we use a screen with a slit ?x.

Because of diffraction, the electron wave spreads

within a cone 2? behind the slit

The uncertainty in the momentum along x

Thus, if we measure x with uncertainty ?x, at the

next moment we wont be able to determine px with

an accuracy better than ?px .

Heisenberg Microscope

The diffraction-limited size of the image of a

point-like object

The scattered photons are collected within an

angle ?

...not a bug, but a feature...

The U.P. is not caused by the measurement

process, it is an intrinsic aspect of the wave

nature of quantons.

It reflects the properties of quantum states

rather then the limitations on the accuracy of

measurements. In contrast to a widespread belief,

the U.P. does not limit accuracy of measurements.

One can perform measurements with an ensemble of

identical particles for some of them, ?x is

measured, for the other - ?p, however, the

dispersions of these measurements would satisfy

the condition (?x2)?(?p2)gth2/4 despite the fact

that only one type of measurements has been done

on each particle.

Q.M. vs. Cl.M.

- to figure out whether one needs to use Q.M. or

Cl.M., we can estimate this product (if it is gtgt

h, the behavior is classical).

Examples

Air at T300K and normal pressure

Distance between the molecules

Cl.M. is okay

Cold atomic gases

Q.M. !

- Bose-Einstein condensation if atoms are boson
- degenerate Fermi gas if atoms are fermions

Problem

A certain device is designed to make a

simultaneous measurement of the position and

velocity of an electron. If the device is

designed to measure the velocity to an accuracy

of one part in 106, what will be the limitation

on the accuracy of the corresponding position

measurement if the velocity measurements yields

the result (a) v2.9?106 m/s, (a) v2.9?108 m/s.

non-relativistic

(a)

(b)

relativistic

Problem

How accurately can the position of a 2.5keV

electron be measured assuming its kinetic energy

is known to 1.

U.P. as a tool to estimate the ground state energy

For a confined motion (bound systems), U.P. sets

a limit how small a momentum (and, thus, the

kinetic energy) could be. This is an important

tool that helps us to estimate the ground-state

energies of various quantum systems.

1. Estimate the lowest possible kinetic energy of

a neutron contained in a typical nucleus of a

radius a 1?10-15m.

non-relativistic

Classical Atom (oxymoron!)

- unambiguous separation of the total energy E

into K and U (we ignore the rest energy)

- the potential energy is the result of

interaction between an electron and the electric

field created by a proton

electrostatic potential (the field characteristic

)

1D motion

2D motion

return point

- no limit on E and, thus, on r !

The Uncertainty Principle applied to an H Atom

- the dependence of the classical energy on the

atoms radius

Quantum approach

minimum of Equantum corresponds to the ground

state the state with a minimum energy

?

non-relativistic motion

The speed of the electron motion in H atom

The dB wavelength of the electron is comparable

with R

semi-classical cartoon (doesnt make sense for

the low-energy states)

Thus, we need Q.M. to describe the system!

The ground state of a quantum harmonic oscillator

(Beiser 3.39) The frequency of oscillation of a

harmonic oscillator of mass m and spring

constant C is

(x its displacement from the equilibrium

position)

The energy of the oscillator is

In classical physics the minimum energy of the

oscillator is 0. Use the U.P. to find Emin in

quantum physics (hint express E in terms of x

only and find Emin from dE/dx0).

Emin - the minimum energy (the ground state

energy)

Zero-Point Oscillations

The zero-point motion persists even at T0.

- the smaller the mass, the larger the amplitude

of zero-point oscillations

Light atoms the zero-point oscillations are

sufficient to prevent liquid Helium-4 from

freezing at atmospheric pressure, no matter how

low the temperature.

Probing the Quantum Limit of Vibrations

The amplitude of the thermal (classical)

vibrations

Quantum oscillations should become observable at

Nanomechanical systems are already approaching

this limit (see the paper by Schwab and Roukes)

Uncertainty Energy ? Time

For a free (non-relativistic) particle

?t the time scale at which a system changes its

energy by ?E

Incorrect to treat ?t as the measurement time

Example the lifetime of a free neutron is 15

min. How uncertain is its energy?

Example Truly monochromatic light corresponds to

an infinitely long plane wave. What is the spread

in the frequency of monochromatic light after

it passes through a fast shutter that forms

1-?s-long light pulses?

Example Calculate how long a virtual

electron-positron pair can exist.

The minimum amount of energy that needs to be

borrowed to make a pair -

By the uncertainty principle, the maximum time

for which such borrowing can go on

HW 4

Homework 4 Beiser Ch. 3, Problems 1, 5, 12,

14, 19, 24, 32, 38

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