2nd half of the Foundations of Nanotechnology Module on introductory quantum
... R. Liboff, Introductory Quantum Mechanics, 4th edition. ▫ D.J. Griffiths ...
Fall 2017 NNSE 507 Quantum 1A
1st half of the Foundations of Nanotechnology Module on introductory quantum mechanics.
Instructor: Yongqiang (Alex) Xue, Associate Professor of Nanoscience Office: CESTM B230C Office Hour: Monday 3-4 PM or by appointment E-mail:
[email protected]
Course Website http://www.albany.edu/~yx152122/Quantum1A-17.html
How to do well in class Study reading materials before AND after Class ALWAYS try to work independently on homework problems
Homework and Grade Homework is assigned each week and due on the following week. Grade: Homework + Class Participation
Course Topics and Texts
Course Topics Fundamental concepts and postulates of QM Operators Superposition principle Commutator relations 1-D systems: Harmonic oscillator
Course Text R. Liboff, Introductory Quantum Mechanics, 4th edition. D.J. Griffiths, Introduction to Quantum Mechanics, 2nd edition.
Course Prerequisites Calculus, Linear Algebra, Differential Equations,… M. Abramowitz and I.A. Stegun, Handbook of Mathematical Functions General classical and modern physics
Week 1 Introduction to Quantum Mechanics
Energy Quanta
Wave-Particle Duality
Review of Classical Waves The double-slit experiment
The Heisenberg Uncertainty Principle
Schrodinger’s Wave Equation
Mathematics of Wavefunction Fourier transforms Wavepackets and uncertainty Operators
Planck’s Energy Quanta: Photoelectric Effects
The photoelectric effect shows the discrete nature of the photon and demonstrates the particle-like behavior of the photon .
Einstein explained the photoelectric effect by postulating that, in a photon, the energy was proportional to the frequency (1905).
Compton Effect In this experiment, an x-ray beam is incident on a solid. A portion of the beam was deflected and the frequency of the deflected wave was shifted compared to the incident wave. The Compton effect, was explained by proposing that photons carry momentum.
If a photon collides with an electron, the wavelength and trajectory of the photon is observed to change. The wavelength shift of light after collision with an electron is consistent with a transfer of momentum from a photon to the electron. The loss of photon energy is reflected in a red shift of its frequency.
De Broglie’s Wave-Particle Duality Principles In 1924. de Broglie postulated the existence of matter waves. He suggested that since waves exhibit particle-like behavior, then particles should also be expected to show wave-like properties.
The hypothesis of de Broglie was the existence of a wave-particle duality principle. The momentum of a photon is given by
Then, de Broglie hypothesized that the wavelength of a particle can be expressed as (de Broglie wavelength of the matter wave).
Davisson-Germer Experiment
Electron Diffraction through Double Slits
Classically, we would predict that electrons passing through slits in a screen should continue in straight lines, forming an exact image of the slits on the rear screen. In practice, however, a series of lines is formed on the rear screen, suggesting that the electrons have been somehow deflected by the slits.
Review of Classical Wave
A wave is a periodic oscillation. It is convenient to describe waves using complex numbers. For example consider the function
where x is position. This function can be plotted on the complex plane as a function of position, x. The phase of the function is the angle on the complex plane.
Review of Classical Wave
The wavelength is defined as the distance between spatial repetitions of the oscillation. This corresponds to a phase change of
This wave is independent of time, and is known as a standing wave.
We could define a function whose phase varies with time:
We define the period, T, as the time between repetitions of the oscillation
Review of Classical Wave
We can combine time and spatial phase oscillations to make a traveling wave.
The intensity of this wave is uniform everywhere it is known as a plane wave.
A plane wave has at least four dimensions (real amplitude, imaginary amplitude, x, and t), so we plot planes of a given phase. These plane waves move through space at the phase velocity of the wave
The Double Slit Experiment Far from the double slit, the electrons from each slit can be described by plane waves, where s is the separation between the slits and L is the distance to the viewing screen. When the planes of constant phase collide, a bright line corresponding to a high intensity of electrons is observed.
At the viewing screen we have
The Double Slit Experiment
At the screen, constructive interference between the plane waves from each slit yields a regular array of bright lines, corresponding to a high intensity of electrons. In between each pair of bright lines, is a dark band where the plane waves interfere destructively, i.e. the waves are out of phase with one another.
The spacing between the bright lines at the viewing screen is
It is notable that the fringe pattern is independent of intensity. Thus, the interference effect should be observed even if just a single electron is fired at the slits at a time.
The only conclusion is that the electron – which we are used to thinking of as a particle - also has wave properties.
The cumulative electron distribution after passage through a double slit. Just a single electron is present in the apparatus at any time.
A. Tanamura et al., Am. J. Phys. 57, 117 (1989).
The Electromagnetic Spectrum
Heisenberg’s Uncertainty Principle
Heisenberg’s uncertainty principle, given in 1927, applies primarily to very small particles, and states that we cannot describe with absolute accuracy the behavior of these subatomic particles.
The uncertainty principle describe a fundamental relationship between conjugate variables, including position and momentum and also energy and time.
The first statement of the uncertainty principle is that it is impossible to simultaneously describe with absolute accuracy the position and momentum of a particle.
The second statement of the uncertainty principle is that it is impossible to simultaneously describe with absolute accuracy the energy of a particle and the instant of time the particle has this energy.
Heisenberg’s Uncertainty Principle
The uncertainty principle is not very relevant to everyday objects
Schrodinger’s Wave Equation The various experimental results involving electromagnetic waves and particles, which could not be explained by classical laws of physics, showed that a revised formulation of mechanics was required.
Erwin Schrodinger (1926), provided a formulation, called wave mechanics, which incorporated the principles of quanta introduced by Planck, and the wave-particle duality principle introduced by de Broglie.
Based on the wave-particle duality principle. we will describe the motion of electrons in a solid or nanostructure by wave theory. This wave theory is described by Schrodinger's wave equation.
Wavefunction of Electrons
Our purposes require a suitable mathematical description for the electron that can describe both its particle and wave-like properties. Following the conventions of quantum mechanics, we will define a function known as the wavefunction to describe the electron. It is typically a complex function and it has the important property that its magnitude squared is the probability density of the electron at a given position and time.
Max Born postulated in 1926 that the function P(x,t)dx is the probability of finding the particle between x and x + dx at a given time t.
If the wavefunction is to describe a single electron, then the sum of its probability density over all space must be 1.
In this case we say that the wavefunction is normalized such that the probability density sums to unity.
Time Domain and Frequency Domain Descriptions of Waves
Consider the wavefunction which describes a wave with amplitude a, intensity and phase oscillating in time at fixed angular frequency This wave carries two pieces of information, its amplitude and angular frequency. Describing the wave in terms of a and is known as the frequency domain description.
Real Space and K-Space Descriptions of Waves
Similarly, consider the wavefunction which describes a wave with amplitude a and phase oscillating in space with spatial frequency or wavenumber . Again, this wave carries two pieces of information, its amplitude and wavenumber. We can describe this wave in terms of its spatial frequencies in k-space, the equivalent of the frequency domain for spatially oscillating waves.
Time Domain and Frequency Domain Descriptions of Waves
Real Space and K-Space Descriptions of Waves
Frequency Domain and K-Space Descriptions of Waves Observe in the above figures that a precise definition of both the position in time and the angular frequency of a wave is impossible.
A wavefunction with angular frequency of precisely is uniformly distributed over all time. Similarly, a wavefunction associated with a precise time t0 contains all angular frequencies.
In real and k-space we also cannot precisely define both the wavenumber and the position.
A wavefunction with a wavenumber of precisely k0 is uniformly distributed over all space. Similarly, a wavefunction localized at a precise position x0 contains all wavenumbers.
Linear Combination of Waves
We plot a wavefunction that could describe an electron that is probable at both position x1 and position x2.
Fourier Transform and Linear Combination of Waves
The
Dirac delta function
Wave Packet and Uncertainty
We now have two ways to describe an electron. We could describe it as a plane wave, with precisely defined wavenumber and angular frequency:
But as we have seen, the intensity/probability density of the plane wave is uniform over all space (and all time). Thus, the position of the electron is perfectly uncertain
On the other hand, we could describe the electron as an idealized point particle existing at a precisely defined position and time.
But the probability density of the point particle is uniform over all of k-space and the frequency domain. We will see that this means the energy and momentum of the electron is perfectly uncertain.
The only alternative is to accept an imprecise description of the electron in both real
space and k-space, time and the frequency domain.
Wave Packet and Uncertainty
A localized oscillation in both representations is called a wave packet. A common wavepacket shape is the Gaussian.
In k-space, the electron is also described by a Gaussian.
(a)
(b)
(c)
the average position and wavenumber of the packet is x=0 and k=0, respectively. The average position has been shifted to =x0. The average wavenumber has been shifted to =k0.
Shift Operation in Real and Inverse Coordinates
Initially the probability distribution is centered at x = 0 and k = 0.
If we shift the wavepacket in k-space to an average value = k0, this is equivalent to multiplying by a phase factor exp[ik0x] in real space.
Similarly, shifting the center of the wavepacket in real space to = x0 is equivalent to multiplying the k-space representation by a phase factor exp[-ikx0].
Expectation Values of Position Given that P(x) is the probability density of the electron at position x, we can determine the average, or expectation value of x from
The Dirac Bra and Ket Notation
Parseval’s Theorem
The expectation value of k is obtained by integrating the wavefunction over all k. This must be performed in k-space.
From Expectation Values to Operators
Summary of Operators
Schrodinger’s Wave Equation
