ECE 487 Lecture 2 : Foundations of Quantum Mechanics...

ECE 487Lecture 2 : Foundations of

Quantum Mechanics I

Class Outline:

Starting to think about quantum mechanics. Introduction to the Schrdinger EquationWaves and Diffraction

What should we expect from quantum mechanics?

What is the Schrdinger equation?

What is diffraction? What is wave-particle duality?

M. J. Gilbert ECE 487 Lecture 2 01 /20/1 1

Things you should know when you leave

Key Questions


Foundations of Quantum Mechanics - ISo, lets start thinking about quantum mechanics

Quantum mechanics is part of everyday life!

We use it to explain the color of an object

Classical mechanics cannot explain the color of anything!

Most colors come from specific absorbing transitions in materials.

All of the transition energies, frequencies, and therefore colors are determined quantum mechanically.

Even the color of the glow of a very hot object cannot be explained by classical physics. We need the quantum statistics of radiation.

This also tells us that quantum mechanics is a crucial part of other sciences, such as chemistry.


We know that quantum mechanics is useful in device design, but can its influence go beyond thatQuantum mechanics is an astonishing intellectual achievement

Arguably the greatest of the twentieth century.

It challenges our prior beliefs about how the world actually works

Some feel that it is never wrong.

It has bizarre but true consequences

Tunneling allows particles to pass through energy barriers that are very large.

We cannot know both the position and momentum of a particle (Heisenbergs Uncertainty Principle).

A particle can exist in a superposition state (neither up or down).

When we measure it, we always find a definite value (Collapse the wavefunction)

Foundations of Quantum Mechanics - I


Quantum mechanics also raises a lot of issues

What do we mean by a measurement of a quantum system?

Schrdingers cat.

Quantum mechanics is non-local.

An event taking place here can apparently instantaneously give a consequence elsewhere.Though this is not apparently useful for anything (things cant be transmitted faster than the speed of light).

It has a statistical nature, but:

It may well be a complete theory unlike classical statistical mechanics which presumes knowledge of position and momentum for all particles.It always provides definite values.It tells us that the world cannot be comprised of purely hidden local variables or things we have not yet seen.



What else?

It continues to provide new surprises in the theory of elementary particles

These discoveries speak directly to the origins of the universe.

It possess strange features which physicists and engineers use for handling and storing information.

It may be able to solve problems that no classical computer can solve by using entangled states.It may be able to create codes that cannot be cracked.



The recipes for using quantum mechanics in a broad range of practicalproblems and engineering designs are relatively straightforward.

though strange and expressed using a different set of ideas and concepts.

If we only ask questions about quantities that can be measured

there are no philosophical problems that prevent us from calculating anything that we could measure.the philosophical approach of only dealing with questions that can be answered by measurement, and regarding all other questions as meaningless, is known as logical positivism.

When we use quantum mechanical principles in tangible applications

e.g., electronic or optical devices and systems.the apparently bizarre aspects become routine



What about performing a calculation?

The mathematical techniques used in quantum mechanics are familiar

Most calculations require:

Performing integralsManipulating matrices.Writing short computer programs.

Many underlying mathematical concepts are quite familiar to engineers

Fourier analysis,Other linear transforms.

The real problem lies in understanding what classical notions have to be discarded and what is the correct understanding that must replace it.



The best and most logical place to start learning quantum mechanics is to examine the Schrdinger equation


In quantum mechanics the Schrdinger equation is one of the most useful in quantum mechanics.

It solves many problems involving quantum mechanical particles that have mass.

This is for non-relativistic particles. In other words, for particles which are moving much slower than the speed of light.

The Schrdinger equation is NOT ideally suited for handling magnetic effects.

The Schrdinger equation is great for exposing many quantum mechanical concepts.

Probability amplitudeLinearityEigenstates


When we consider the Schrdinger equation, lets start with the simplest casethe time-independent case.


Electrons are very strange animals because the behave both as waves and as individual particles. This is known as wave-particle duality. How can we see this behavior?

We can see electrons behaving as waves if we place a piece of metal in a large electric field in a vacuum.

Field couples to the electrons and pulls them out of the metal.The acceleration they feel in the fixed potential gives them the same kinetic energy.

We know electrons behave like particles, when we take the beam of electrons and shine it on the surface of a crystal with the scattered electrons landing on a phosphorescent screen.

Beam of electrons forms a diffraction pattern of dots on the screen.They look like they have a definite charge and mass.


So if electrons have wavelike properties, then we can apply some of our intuition about waves to them to understand how the move


Electrons should definitely have a wavelength:

Where p is the electron momentum and h is Plancks constant:

This is known as de Broglies hypothesis.

We can use what we know from electromagnetics to describe how the electron moves through space:

Electromagnetics also gives us a prescription to describe how this wave evolves in space through the Helmholtz wave equation for a monochromatic wave:

In one dimension


Now we have an electron wave and a method to describe how is moves in position, lets put them together


We get a series of possible solutions:

We should note that things dont really change if we move from 1D systems to 3D systems:

where Laplacianoperator

The solutions are still the same as in the 1D case but now both r and k are vectors.

As well as their negative counterparts.


Now lets analyze the Helmholtz equation further


Generally, for our Helmholtz equation we may define the wavevector magnitude, k, as:

Or given the empirical wavelength that the electrons exhibit:

where

With this expression, we can plug it back into our original Helmholtz wave equation:

But remember the electron has mass, so lets divide both sides by twice this mass:


But we should not forget completely the lessons we learned from classical Newtonian mechanics


Where v is the velocity. Kinetic energy of the electron

then

And in general

So, lets rewrite our wave equation

We can rearrange this yet again to get the more common form:

Time independent Schrdinger equation for a particle of mass, m0.


So lets ask some very simple questions


NO! All that we have done is merely used the wavelike nature of the electron incombination with classical electromagnetics to suggest it as an equation that agreeswith at least one experiment.

Question # 1: Dude, did we just derive Schrdingers equation?

Question # 2: Bummer, can we derive Schrdingers equation?

NO! There are no first principles in the physics that preceded quantum mechanics that predicts anything like such wave behavior for the electron.

Schrdingers equation has to be postulated just like Newton did for his laws of motion.

The ONLY justification for using this equation is that it works!!


When we apply quantum mechanics to a problem, we will want to know something about where this lone electron is located in our system


To obtain this information, we use probability densities.

In practice, we fine that the probability, P(r), of finding an electron near any specific point r in space is

Proportional to the modulus squared Of the wave

Some things to know about the squared modulus :

It assures that we will always have a positive quantity. This is good because who knows what negative probability means?

It is consistent with other uses of the squared modulus. For example, squared amplitude tells us the energy density or intensity in a wave motion of an electromagnetic wave.

Here the modulus squared of the wave amplitude gives the wave intensity or the probability of finding an electron at any point in space


So, how should we refer to this quantity that we are examining?


Since the probability is given by the squared modulus of the wavefunction, we then refer to the wavefunction as a probability amplitude or quantum mechanical amplitude

Be sure to be careful as this probability amplitude is very different from the probability itself!

The probability amplitude has little or no precedent in classical physics or classical statistics.

For now, lets think of the probability amplitude as being the amplitude of a wave.

Later we will find that the concept of probability amplitude extends into quite different descriptions but still retains the concept of the modulus squared representing a probability.


Lets consider an example where we can see both the wavelike properties of the electron and the particle like properties.


We can do this by examining simple electron diffraction, or in optics, Youngs double slit experiment. Consider two very narrow slits separated by a distance, s.

Narrow in relation to the wavelength too!


Now turn on a monochromatic light source, like a laser and shine the beam of electrons, each with wavevector k, directly at the screen


To make our analysis simpler, assume that the screen is far away from the slits, z0 >> s.

Each one of the openings can be considered asource of circularly expanding waves (HuygensPrinciple).

We can now construct the wavefunction for the net wave arriving at the screen:

Where we have used the fact that in the xz plane, a circle of radius r centered at x = a and z = 0 is: ( ) 222 zaxr +=


Lets presume that we only care about small angles, x


To connect this back to our initial picture of how the fringes should behave we need to compute the probability amplitude of our total wavefunction:


Hence, a beam ofmonoenergetic electronsproduces a cosinusoidalinterference pattern onthe screen with fringesseparated by ds = z0 /s !

http://upload.wikimedia.org/wikipedia/commons/6/63/Wavefronts_of_two_coherent_point_sources.png


These interference effects are quite strange and cannot be understood classically


Lets block one of the slits.

We see no fringes.

Now unblock the slit.

We see fringes. Areas once bright are now dark.

How can we possibly explain that opening a secondsource of particles actually reduces the number ofparticles arriving at some point on the screen?

http://upload.wikimedia.org/wikipedia/commons/3/3c/Wave_Diffraction_4Lambda_Slit.pnghttp://upload.wikimedia.org/wikipedia/commons/f/f8/Double_slit_diffraction.svg


We can try a something intuitive


Particles from the second slit are bouncing off particles from the first slit.They avoid some parts of the screen because of the collisions.

Repeat the experiment with extremely low electron current so only one electron is in the apparatus at a time.

Same pattern emerges.

Must use interference amplitudes to explain results of experiment.

Waves explain the physics.

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ECE 487 Lecture 2 : Foundations of Quantum Mechanics...

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