The fall of Classical Physics - Uniuddeangeli/fismod/Fisica3qm.pdf · The fall of Classical Physics...

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The fall of Classical Physics

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Classical physics: Fundamental Models

Particle Model (particles, bodies)

Motion in 3 dimension; for each time t, position and speed

are known (they are well-defined numbers, regardless we

know them). Mass is known.

Systems and rigid objects

Extension of particle model

Wave Model (light, sound, …)

Generalization of the particle model: energy is transported,

which can be spread (de-localized)

Interference

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Classical physics at the end of XIX Century

Scientists are convinced that the particle and wave model can describe the evolution of the Universe, when folded with

Newton’s laws (dynamics)

Description of forces Maxwell’s equations

Law of gravity.

…

We live in a 3-d world, and motion happens in an absolute time. Time and space (distances) intervals are absolute.

The Universe is homogeneous and isotropical; time is homogeneous. Relativity

The physics entities can be described either in the particle or in the wave model.

Natura non facit saltus (the variables involved in the description are continuous).

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Something is wrong

Relativity, continuity, wave/particle (I)

Maxwell equations are

not relativistically

covariant!

Moreover, a series of

experiments seems to

indicate that the

speed of light is

constant (Michelson-

Morley, …) A speed!

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Something is wrong

Relativity, continuity, wave/particle (IIa)

In the beginning of the XX

century, it was known that

atoms were made of a heavy nucleus, with

positive charge, and by

light negative electrons

Electrostatics like gravity:

planetary model

All orbits allowed

But: electrons, being

accelerated, should radiate

and eventually fall into the

nucleus

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Something is wrong

Relativity, continuity, wave/particle (IIb)

If atoms emit energy in the form of photons due to

level transitions, and if color is a measure of energy,

they should emit at all wavelengths – but they don’t

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Something is wrong

Relativity, continuity, wave/particle (III)

Radiation has a particle-like behaviour, sometimes

Particles display a wave-like behaviour, sometimes

=> In summary, something wrong involving the

foundations:

Relativity

Continuity

Wave/Particle duality

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Need for a new physics

A reformulation of physics was needed

This is fascinating!!! Involved philosophy, logics, contacts

with civilizations far away from us…

A charming story in the evolution of mankind

But… just a moment… I leaved up to now with classical

physics, and nothing bad happened to me!

Because classical physics fails at very small scales, comparable with

the atom’s dimensions, 10-10 m, or at speeds comparable with the

speed of light, c ~ 3 108 m/s

Under usual conditions, classical physics makes a good job.

Warning: What follows is logically correct, although

sometimes historically inappropriate.

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I

Light behaves like a particle,

sometimes

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Photoelectric Effect Features

and Photon Model explanation

The experimental results contradict all four classical predictions

Einstein interpretation: All electromagnetic radiation can be considered a stream of quanta, called photons

A photon of incident light gives all its energy hƒ to a single electron in the metal

h is called the Planck constant, and plays a fundamental role in Quantum Physics

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The Compton Effect

Compton dealt with Einstein’s idea of

photon momentum

Einstein: a photon with energy E carries a momentum of E/c = hƒ / c

According to the classical theory,

electromagnetic waves of frequency ƒo

incident on electrons should scatter,

keeping the same frequency – they scatter the electron as well…

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Compton’s experiment showed that, at

any given angle, a different frequency of

radiation is observed

The graphs show the scattered x-ray for

various angles

Again, treating the photon as a particle of

energy hf explains the phenomenon. The

shifted peak, ‘> 0, is caused by the scattering of free electrons

This is called the Compton shift equation

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Compton Effect, Explanation

The results could be explained, again, by treating the photons as point-like particles having

energy hƒ

momentum hƒ / c

Assume the energy and momentum of the isolated system of the colliding photon-electron are conserved

Adopted a particle model for a well-known wave

The unshifted wavelength, o, is caused by x-rays scattered from the electrons that are tightly bound to the target atoms

The shifted peak, ', is caused by x-rays scattered from free electrons in the target

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Every object at T > 0 radiates electromagnetically, and

absorbes radiation as well

Stefan-Boltzmann law:

Blackbody: the

perfect absorber/emitter

Blackbody radiation

“Black” body

Classical interpretation: atoms in the object vibrate; since <E> ~

kT, the hotter the object, the more energetic the vibration, the

higher the frequency

The nature of the radiation leaving the cavity through the

hole depends only on the temperature of the cavity walls

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Experimental findings

& classical calculation

Wien’s law: the emission

peaks at

Example: for Sun T ~ 6000K

But the classical calculation

(Rayleigh-Jeans) gives a

completely different result…

Ultraviolet catastrophe

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Experimental findings

& classical calculation

Classical calculation (Raileigh-

Jeans): the blackbody is a set of

oscillators which can absorb any

frequency, and in level transition

emit/absorb quanta of energy:

No maximum; a ultraviolet

catastrophe should absorb all

energy Experiment

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Planck’s hypothesis

Only the oscillation modes for which

E = hf

are allowed…

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Interpretation

The classical calculation is accurate for large

wavelengths, and is the limit for h -> 0

Elementary oscillators can have only

quantized energies, which satisfy

E=nhf (h is an universal constant, n is an integer –quantum- number)

Transitions are accompanied by the

emission of quanta of energy (photons)

n

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3

2

1

E

4hf

3hf

2hf

hf

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Which lamp emits e.m. radiation ?

1) A

2) B

3) A & B

4) None

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Particle-like behavior of light:

now smoking guns… The reaction

has been

recorded

millions of times…

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Bremsstrahlung

"Bremsstrahlung" means in German "braking radiation“; it is the radiation emitted when electrons are decelerated or "braked" when they are fired at a metal target. Accelerated charges give off electromagnetic radiation, and when the energy of the bombarding electrons is high enough, that radiation is in the x-ray region of the electromagnetic spectrum. It is characterized by a continuous distribution of radiation which becomes more intense and shifts toward higher frequencies when the energy of the bombarding electrons is increased.

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Summary

The wave model cannot explain the behavior of light

in certain conditions

Photoelectric effect

Compton effect

Blackbody radiation

Gamma conversion/Bremsstrahlung

Light behaves like a particle, and has to be

considered in some conditions as made by single

particles (photons) each with energy

h ~ 6.6 10-34 Js is called the Planck’s constant

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II

Particles behave like waves,

sometimes

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Should, symmetrically, particles display

radiation-like properties?

The key is a diffraction experiment: do particles show

interference?

A small cloud of Ne atoms was cooled down to T~0. It

was then released and fell with zero initial velocity onto

a plate pierced with two parallel slits of width 2 μm, separated by a distance of d=6 μm. The plate was

located H=3.5 cm below the center of the laser trap.

The atoms were detected when they reached a screen located D=85 cm below the plane of the two slits. This

screen registered the impacts of the atoms: each dot represents a single impact. The distance between two

maxima, y, is 1mm.

The diffraction pattern is consistent with the diffraction of waves with

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Diffraction of electrons

Davisson & Germer 1925:

Electrons display diffraction patterns !!!

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de Broglie’s wavelength

What is the wavelength associated to a particle?

de Broglie’s wavelength:

Explains quantitatively the diffraction by Davisson and Germer……

Note the symmetry

What is the wavelength of an electron moving at 107 m/s ?

(smaller than an atomic length; note the dependence on m)

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Atomic spectra

Why atoms emit according to a discrete energy spectrum?

Something must

be there...

Balmer

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Electrons in atoms: a semiclassical model

Similar to waves on a cord, let’s imagine that

the only possible stable waves are stationary…

2 r = n n=1,2,3,…

=> Angular momentum is quantized (Bohr

postulated it…)

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v

r

m

F

NB:

• In SI, ke = (1/4 0) ~ 9 x 109 SI units

• Total energy < 0 (bound state)

• <Ek> = -<Ep/2> (true in general for bound states, virial theorem)

Only special values are possible for the radius !

Hydrogen (Z=1)

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Energy levels

The radius can only assume

values

The smallest radius (Bohr’s radius) is

Radius and energy are related:

And thus energy is quantized:

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Transitions

An electron, passing from an orbit of energy Ei

to an orbit with Ef < Ei, emits energy [a photon

such that f = (Ei-Ef)/h]

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Level transitions and energy quanta

We obtain Balmer’s relation!

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Limitations

Semiclassical models wave-particle duality can explain

phenomena, but the thing is still insatisfactory,

When do particles behave as particles, when do they behave

as waves?

Why is the atom stable, contrary to Maxwell’s equations?

We need to rewrite the fundamental models, rebuilding

the foundations of physics…

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Wavefunction

Change the basic model!

We can describe the position of a particle

through a wavefunction (r,t). This can account

for the concepts of wave and particle (extension

and simplification).

Can we simply use the D’Alembert waves, real

waves? No…

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Wavefunction - II

We want a new kind of “waves” which can account for particles, old waves, and obey to F=ma.

And they should reproduce the characteristics of “real” particles: a particle can display interference corresponding to a size of 10-7 m, but have a radius smaller than 10-10 m

Waves of what, then? No more of energy,

but of probability

The square of the wavefunction is the intensity, and it gives the probability to find the particle in a given time in a given place.

Waves such that F=ma? We’ll see that they cannot be a function in R, but that C is the minimum space needed for the model.

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SUMMARY Close to the beginning of the XX century, people thought that

physics was understood. Two models (waves, particles). But:

Quantization at atomic level became experimentally evident

Particle-like behavior of radiation: radiation can be considered in some

conditions as a set of particles (photons) each with energy

Wave-like property of particles: particles behave in certain condistions as

waves with wavenumber

Role of Planck’s constant, h ~ 6.6 10-34 Js

Concepts of wave and particle need to be unified: wavefunction

(r,t).

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L’equazione di Schroedinger

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Proprieta’ della funzione d’onda

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L’equazione di S.

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L1 – The Schroedinger

equation • A particle of mass m on the x-axis is subject to

a force F(x, t)

• The program of classical mechanics is to

determine the position of the particle at any

given time: x(t). Once we know that, we can

figure out the velocity v = dx/dt, the momentum

p = mv, the kinetic energy T = (1/2)mv2, or any

other dynamical variable of interest.

• How to determine x(t) ? Newton's second law:

F = ma.

– For conservative systems - the only kind that occur

at microscopic level - the force can be expressed as

the derivative of a potential energy function, F = -dV/

dx, and Newton's law reads m d2x/dt2 = -dV/ dx

– This, together with appropriate initial conditions

(typically the position and velocity at t = 0),

determines x(t).

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• A particle of mass m, moving along the

x axis, is subject to a force

F(x, t) = -dV/ dx

• Quantum mechanics approaches this

same problem quite differently. In this

case what we're looking for is the wave

function, (x, t), of the particle, and we

get it by solving the Schroedinger

equation:

• In 3 dimensions,

~ 10-34 Js

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The statistical interpretation

• What is this "wave function", and what can it tell you? After all, a particle, by its nature, is localized at a point, whereas the wave function is spread out in space (it's a function of x, for any given time t). How can such an object be said to describe the state of a particle?

• Born's statistical interpretation:

Quite likely to find the particle near A, and relatively unlikely near B.

• The statistical interpretation introduces a kind of indeterminacy into quantum mechanics, for even if you know everything the theory has to tell you about the particle (its wave function), you cannot predict with certainty the outcome of a simple experiment to measure its position

– all quantum mechanics gives is statistical information about the possible results

• This indeterminacy has been profoundly disturbing

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Realism, ortodoxy,

agnosticism - 1

• Suppose I measure the position of the particle,

and I find C. Question: Where was the particle

just before I made the measurement?

There seem to be three plausible answers to

this question…

1. The realist position: The particle was at C. This

seems a sensible response, and it is the one

Einstein advocated. However, if this is true QM

is an incomplete theory, since the particle

really was at C, and yet QM was unable to tell

us so. The position of the particle was never indeterminate, but was merely unknown to the

experimenter. Evidently is not the whole

story: some additional information (a hidden

variable) is needed to provide a complete

description of the particle

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Realism, ortodoxy,

agnosticism - 2




2. The orthodox position: The particle wasn't

really anywhere. It was the act of

measurement that forced the particle to "take a

stand“. Observations not only disturb what is to be measured, they produce it .... We compel

the particle to assume a definite position. This

view (the so-called Copenhagen interpretation)

is associated with Bohr and his followers.

Among physicists it has always been the most widely accepted position. Note, however, that

if it is correct there is something very peculiar

about the act of measurement - something that

over half a century of debate has done

precious little to illuminate.

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Realism, ortodoxy,

agnosticism - 3




3. The agnostic position: Refuse to answer. This

is not as silly as it sounds - after all, what

sense can there be in making assertions about

the status of a particle before a measurement, when the only way of knowing whether you

were right is precisely to conduct a

measurement, in which case what you get is

no longer "before the measurement"? It is

metaphysics to worry about something that cannot, by its nature, be tested. One should

not think about the problem of whether

something one cannot know anything about

exists

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Realism, ortodoxy,

or agnosticism?

• Suppose I measure the position of the particle, and I find C. Question: Where was the particle just before I made the measurement?

• Until recently, all three positions had their partisans. But in 1964 John Bell demonstrated that it makes an observable difference if the particle had a precise (though unknown) position prior to the measurement. Bell's theorem made it an experimental question whether 1 or 2 is correct. The experiments have confirmed the orthodox interpretation: a particle does not have a precise position prior to measurement; it is the measurement that insists on one particular number, and in a sense creates the specific result, statistically guided by the wave function.

• Still some agnosticism is tolerated…

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Collapse of

the

wavefunction

• Suppose I measure the position of the particle, and I find C. Question: Where will be the particle immediately after?

• Of course in C. How does the orthodox interpretation explain that the second measurement is bound to give the value C? Evidently the first measurement radically alters the wave function, so that it is now sharply peaked about C. The wave function collapses upon measurement (but soon spreads out again, following the Schroedinger equation, so the second measurement must be made quickly). There are, then, two entirely distinct kinds of physical processes: "ordinary", in which evolves under the Schroedinger equation, and "measurements", in which suddenly collapses.

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Normalization

• | (x, t)|2 is the probability density for finding

the particle at point x at time t.

The integral of | (x, t)|2 over space must be 1

(the particle has to be somewhere).

• The wave function is supposed to be

determined by the Schroedinger equation--we can't impose an extraneous condition on

without checking that the two are consistent.

• Fortunately, the Schroedinger equation is

linear: if is a solution, so too is A , where

A is any (complex) constant. What we must

do, then, is pick this undetermined

multiplicative factor so that The integral of |

(x, t)|2 over space must be 1 This process is called normalizing the wave function.

• Physically realizable states correspond to the

"square-integrable" solutions to Schroedinger's

equation.

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Will a normalized function stay as such?

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Expectation values

• For a particle in state , the expectation value

of x is

• It does not mean that if you measure the

position of one particle over and over again,

this is the average of the results

– On the contrary, the first measurement (whose

outcome is indeterminate) will collapse the wave

function to a spike at the value obtained, and the

subsequent measurements (if they're performed

quickly) will simply repeat that same result.

• Rather, <x> is the average of measurements

performed on particles all in the state , which

means that either you must find some way of returning the particle to its original state after

each measurement, or you prepare an

ensemble of particles, each in the same state

, and measure the positions of all of them:

<x> is the average of them.

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Momentum

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More on operators

• One could also simply observe that

Schroedinger’s equations works as if

(exercise: apply on the plane wave). In

3 dimensions,

Compound operators

• Kinetic energy is

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Angular momentum

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Exercise

• A particle is represented at t=0 by the wavefunction

(x, 0) = A(a2-x2) |x| < a (a>0).

= 0 elsewhere

a Determine the normalization constant A

b, c What is the expectation value for x and for p at t=0?

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Exercise (cont.)

• A particle is represented at t=0 by the wavefunction

(x, 0) = A(a2-x2) |x| < a (a>0).

= 0 elsewhere

d, e Compute <x2>, <p2>

f, g Compute the uncertainty on x, p

h Verify the uncertainty principle in this case

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L2 – The time-independent

Schroedinger equation

• Supponiamo che il potenziale U sia

indipendente dal tempo

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, soluzione della prima equazione (eq.agli autovalori detta equazione di S. stazionaria), e’ detta autofunzione

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1. They are stationary states. Although

the wave function itself

does (obviously) depend on t, the

probability density does not - the time

dependence cancels out. The same

thing happens in calculating the

expectation value of any dynamical

variable

3 comments on the stationary solutions: 1

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2. They are states of definite energy. In

mechanics, the total energy is called the

Hamiltonian:

H(x, p) = p2/2m + V(x).

The corresponding Hamiltonian operator,

obtained by the substitution p -> p operator

Note: it is true in general that, if is

eigenfunction of an operator, the measurement

gives as a result certainly the eigenvalue


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3. They are a basis. The general solution

is a linear combination of separable

solutions. The time-independent

Schroedinger equation might yield an

infinite collection of solutions, each with

its associated value of the separation

constant; thus there is a different wave

function for each allowed energy.

The S. equation is linear: a linear

combination of solutions is itself a

solution.

It so happens that every solution to the

(time-dependent) S. equation can be

written as a linear combination of

stationary solutions.

To really play the game, mow we must

input some values for V


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Free particle (V=0, everywhere)

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The infinite square well

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Infinite square well, 2

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But…

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11/6/12

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Intermezzo: Heisenberg principle (theorem)

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11/6/12

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The uncertainty principle (qualitative)

Imagine that you're holding one end of a long rope, and you generate a

wave by shaking it up and down rhythmically.

Where is that wave? Nowhere, precisely - spread out over 50 m or so.

What is its wavelength? It looks like ~6 m

By contrast, if you gave the rope a sudden jerk you'd get a relatively narrow

bump traveling down the line. This time the first question (Where precisely is

the wave?) is a sensible one, and the second (What is its wavelength?)

seems difficult - it isn't even vaguely periodic, so how can you assign a

wavelength to it?

=> Uncertainty is a characteristic of the wave representation

x(m)

x(m)

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The more precise a wave's x is, the less precise is , and vice versa. A theorem in Fourier analysis makes this rigorous…

This applies to any wave, and in particular to the QM wave function. is related to p by the de Broglie formula

Thus a spread in corresponds to a spread in p, and our observation says that the more precisely determined a particle's position is, the less precisely is p

This is Heisenberg's famous uncertainty principle. (we'll prove it later, but I want to anticipate it now)

The uncertainty principle (qualitative, II)

x(m)

x(m)

11/6/12

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General rules for the linear piece-wise potential

Divide the interval in N regions with constant

potential Vi

For each region, solve the Schroedinger equation

Real exponentials for E <Vi; imaginary otherwise

Impose boundary/initial conditions

Impose continuity conditions

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Finite Square-Well Potential

The finite square-well potential is

The Schrödinger equation outside the finite well in regions I and III is

or using

yields . Considering that the wave function must be zero at

infinity, the solutions for this equation are

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Inside the square well, where the potential V is zero, the wave equation

becomes where

Instead of a sinusoidal solution we have

The boundary conditions require that

and the wave function must be smooth where the regions meet.

Note that the

wave function is

nonzero outside of the box.

Finite Square-Well Solution

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Penetration Depth

The penetration depth is the distance outside the potential well where

the probability significantly decreases. It is given by

It should not be surprising to find that the penetration distance that

violates classical physics is proportional to Planck’s constant.

E t (V0 E)m x

p= (V0 E)

m

2m(V0 E) 2m(V0 E) 2

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Barriers and Tunneling

Consider a particle of energy E approaching a potential barrier of height V0 and

the potential everywhere else is zero.

We will first consider the case when the energy is greater than the potential barrier.

In regions I and III the wave numbers are:

In the barrier region we have

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Reflection and Transmission

The wave function will consist of an incident wave, a reflected wave, and a

transmitted wave.

The potentials and the Schrödinger wave equation for the three regions are

as follows:

The corresponding solutions are:

As the wave moves from left to right, we can simplify the wave functions to:

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Probability of Reflection and Transmission

The probability of the particles being reflected R or transmitted T is:

Because the particles must be either reflected or transmitted we have:

R + T = 1.

By applying the boundary conditions x ± , x = 0, and x = L, we arrive

at the transmission probability:

Notice that there is a situation in which the transmission probability is 1.

11/6/12

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Tunneling

Now we consider the situation where classically the particle does not have enough energy to surmount the potential barrier, E < V0.

The quantum mechanical result is one of the most remarkable features of modern physics, and there is ample experimental proof of its existence. There is a small, but finite, probability that the particle can penetrate the barrier and even emerge on the other side.

The wave function in region II becomes

The transmission probability that describes the phenomenon of tunneling is

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Uncertainty Explanation

Consider when L >> 1 then the transmission probability becomes:

This violation allowed by the uncertainty principle is equal to the

negative kinetic energy required! The particle is allowed by quantum

mechanics and the uncertainty principle to penetrate into a classically

forbidden region. The minimum such kinetic energy is:

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QM in 3 dimensions

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V(r): separation of variables

3

ˆ = 2,

2 ,

ˆ 2 Y = l l 1Y

ˆ 2 = l l 1

ˆ

2.

4

The angular equation

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The angular equation -

ˆ = i ,

ˆ = m ˆ = m

ˆ

.

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The angular equation -

Pl are the Legendre polynomials, defined by the Rodrigues formula:

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ˆ = l(l +1)

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Spherical harmonics

= (-1)m for m>=0 and =1 for m<0. The Y are orthogonal, so

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The radial equation

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The H atom

u(r) = rR(r)

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Asymptotic behavior

u(r) = rR(r)

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The radial solution

u(r) = rR(r)

n > l

is the q-th Laguerre polynomial.

Remember: only the solutions for n>l are valid functions

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The radial

solution: energy

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Ground state

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n > 1

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n > 1 (continued)

Since they are eigenvectors for different eigenvalues

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Example

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Graphs

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Angular momentum

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