# Electron Spin

In filling the k-space of a metal with electrons, each grid point in the k-space can be occupied with two electrons, one in a spin up state and one in a spin down state. Energy of the electron does not depend on the spin state of the electron. The full quantum state of the electron then includes the position wavefunction and the spin orientation. This can be notates as follows:

The spin of a particle can also be represented as a two-element matrix or spinor, with spin up represented as (1;0) and spin down as (0;1).

http://hyperphysics.phy-astr.gsu.edu/hbase/spin.html

# The Sommerfeld Model – Metals

Following the Drude model describing the movement of electrons in metals, Sommerfeld developed yet another model for electrons in metals in 1927. This new model would account for electron energy distributions in metals, Pauli’s exclusion principle, and Fermi-Dirac statistics of electrons. This model factors in quantum mechanics and the Schrödinger Equation.

The Somerfeld model’s view of electrons in metals can be taken as an example of a large volume of metal with electrons confined in the volume. We’ll call this a potential well. Inside this volume or potential well, electrons are ‘free’ with zero potential. Outside the potential well, the potential is infinite. The electron states inside this box are governed by the Schroedinger equation.

The quantum state of an electron is generally described by the Schroedinger equation as shown below.

The result of the Schroedinger equation, applying the boundary conditions of the problem with the potential being zero at the boundary, the solution of the wavefunction is below. This solution introduces a concept called the k-space, a 3D grid of allowed quantum states.

The density of grid points in the k-space is related to Lx, Ly, and Lz of the solution. The number of grid points per unit volume or density (i.e. density of states) will be V/(2*pi)^3, where the spacing of points in the 3D k-space are defined as 2pi/Lx, 2pi/Ly, and 2pi/Lz.

# Mobility and Saturation Velocity in Semiconductors

In solid state physics, mobility describes how quickly a charge carrier can move within a semiconductor device when in the presence of a force (electric field). When an electric field is applied, the particles begin to move at a certain drift velocity, given by the mobility of the carrier (electron or hole) and electric field. The equation can be written as:

This is also related to Ohm’s law in point form, which is the conductivity multiplied by the Electric field. This shows that the conductivity of a material is related to the number of charge carriers as well as their mobility within the material. Mobility is heavily dependent on doping, which introduces defects to the material. This means that intrinsic semiconductor material (Si or Ge) has higher mobility, but this is a paradox due to the fact that intrinsic semiconductor has no charge carriers. In addition, mobility is inversely proportional to mass, so a heavier particles will move at a slower rate.

Phonons also contribute to a loss of mobility due to an effect known as “Lattice Scattering”. When the temperature of semiconductor material is raised above absolute zero, the atoms vibrate and create phonons. The higher the temperature, the more phonon particles which means greater collisions and lower mobility.

Saturation velocity refers to the maximum velocity a charge carrier can travel within a semiconductor in the presence of a strong electric field. As previously stated, the velocity is proportional to mobility, but with increasing electric field there reaches a point where the velocity saturates. From this point, increasing the field only leads to more collisions with the lattice structure and phonons, which does not help the drift speed. Different semiconductor materials have different saturation velocities and are strong functions of impurities.

# Quantum Wells in LEDs

Previously, the topic of a quantum well’s functionality was discussed. Here, the topic of quantum wells’ function specifically within Light Emitting Diodes is discussed. In fact, quantum wells often implement multiple quantum wells to increase their luminescence, or total light emission.

Quantum wells are formed when a type of semiconductor (or compound semiconductor) with a more narrow bandgap between its conduction and valence band is placed in between two wider bandgap semiconductors (such as GaN or AlN). The quantum well traps electrons within it at the conduction band, so as to increase recombination. Holes from the valence band will recombine with the conduction band electrons to emit photons which gives the LED its distinct emission of light. The quantum well is the reason why the LED does not function strictly as a diode. If the electrons were not trapped, the current would simply flow normally as in a regular LED. Although a greater number of quantum wells increases the luminescence of the LED, it can also lead to defects in the device.

LEDs generate different colors of light by using different semiconductor material and different amounts of doping. This changes the energy gaps and leads to a different wavelength of light being produced. Gallium is a common element used in these compound materials.

# Hermitian Operators, Time-Shifting Wavefunction

It was mentioned in the previous article on Quantum Mechanics [link] that if the integral of a wavefunction over all space at one time is equal to one (thereby meaning that it is normalized and that the probabilility of the particle existing is 100%), then the wavefunction is applicable to a later time, t.

A function in the place of Ψ*Ψ is used as a probability density function, ρ(x,t). The function N(t) is the resultant probability at a given time, given that the probability was found to be equal to 1 at a given time t0. Shown below, it is proposed that for dN/dt to equal zero, the Hamiltonian must be a Hermitian operator.

A Hermitian operator would satisfy the following:

Hermiticity in general may referred to as a type of conjugate form of an operator. An operator is hermitian if the hermitian conjugate is equal to itself. One may compare this relationship as to a real number whose complex conjugate is equal to itself.

Returning to the calculation of dN/dt,

# Ψ Wavefunction Describes Probability

Schroedinger’s first interpretation of the wavefunction was that Ψ would describe how a particle dissipates. Where the wavefunction Ψ was the highest, then that was where more of the particle was present. Max Born disagreed saying that a particle would not dissintegrates, choosing another direction to move. Max Born proposed that the wavefunction would actually describe the probability of a particle inhabiting a space. Both Schroedinger and Einstein were initially opposed to the idea of a probabilistic interpretation of the Schroedinger equation. The probabilistic interpretation of Max Born however later became the consensus view of quantum mechanics.

The wavefunction Ψ therefore describes the probability of finding a particle at position x at time t, not the amount of the particle that exists there.

Since the Schroedinger equation is both a function of position and time, it can only be solved for one variable at a time. Solving for position is preferable due to the fact that if the wavefunction is known for all x, this can provide information for how the wavefunction is at a later time.

Of the limits regarding the wavefunction, it is also said that the wavefunction must be convergent. The wavefunction therefore does not approach a finite constant as x approaches infinity.

We also recall that a wavefunction may also be multiplied by a number. It would appear that doing so would violate the above expression. The answer regarding this conjecture is that the above formula represents a normalized wavefunction. Yet it turns out that not all wavefunctions are normalizable. The case of multiplying the wavefunction with a magnitude in fact would still be normalizable, however. A wavefunction can be normalized if the integral is a finite number less than infinity using the following method:

# Matrices, Multiple Dimensions in Quantum Mechanics

There comes to be two main approaches to Quantum Mechanics. One approach is an equations approach which uses wavefunctions, operators and sometimes eigenstages. The other approach is a linear algebra approach that uses matrices, vectors and eigenvectors to describe quantum mechanics.

Consider an example of a quantum mechanical problem that uses linear algebra for the description of particle spin:

This allows for a more direct view of commutators as discussed in the previous article on quantum mechanics [link]. Matrices have an advantage of storing much more information elegantly and are convenient for commutations.

Matrices in fact can be written for x_hat, p_hat and other operators. Matrices are also useful for introducing more than one dimension. We can also make use of this method to give us a three-dimensional Schroedinger equation. First we will start by forming three dimensions of momentum p vectors.

# Operators in Quantum Mechanics

Before getting into problems relating to the free particle schroedinger equation, let’s review the full Schroedinger equation. The energy operator E_hat appears in the first equation below. Thus far, the euqation relates only kinetic energy. Potential energy however when considered would allow the Schroedinger equation to be applied in a wide range of possible applications, being able to describe the interactions of atoms and molecules and their interactions in free space, wells, and other environments due to the linearity of quantum mechanics. One major point to take from discovering the free particle Schroedinger equation is how important it is in Quantum Mechanics to create energy operators. An operator can be as simple as a constant or as complicated as a partial differential. By allowing an ‘operator’ to take on this wider range of features as opposed to a basic variable makes for the basis of many quantum mechanical calculations. It then follows that the portential, V(x,t) can also be treated as an operator that modifies the system.

Consider an operator X_hat that when multiplied by a function, results in the function being multiplied by x. Remember that although this may look like a variable, it is useful to consider this as an operator in Quantum mechanics.

Does the order in which operators are multiplied matter?

Considering that operators are not always constants or variables, but also sometimes differentials, the order of operations for operators does matter.

A communtator is understood as the difference of linear operators. The commutator of x_hat and p_hat is i*h_bar.

# Direct-Bandgap & Indirect-Bandgap Semiconductors

Direct Semiconductors

When light reaches a semiconductor, the light is absorbed if the photon energy is greater than or equal to the band gap, creating electron-hole pairs. In a direct semiconductor, the minimum of the conduction band is aligned with the maximum of the valence band.

One example of a direct semiconductor is GaAs. The band diagram for GaAs is shown to

the right. As the gap between the valence band and conduction band is 1.42eV, if a

photon of same or greater energy is applied to the semiconductor, a hole-electron pair is created for each photon. This is termed the photo-excitation of semiconductors. The photon is thereby absorbed into the semiconductor.

Indirect Semiconductors and Phonons

For an indirect semiconductor to absorb a photon, the process must be mediated by phonons, which are quanta of sound and in this case refer to the acoustic vibration of crystal lattice. A phonon is also used to provide energy for radiative recombination. When understanding the essence of a phonon, one should recall that sound is not necessarily within hearing range (20 – 20kHz). In fact, the sound vibrations in a semiconductor may well be in the Terrahertz range. The diagram to the right shows how an indirect semiconductor band would appear and also the use of phonon energy to mediate the process of allowing the indirect semiconductor to behave as a semiconductor.

Excitons

Excitons are bound electron-hole pairs that are created in pure semiconductors when a photon with bandgap energy or larger is absorbed. In bulk semiconductors, these excitons will dissipate rapidly. In quantum wells however, the excitons may remain, even at room temperature. The effect of the quantum well is to force an electron and hole to be very close to each other. This allows for a strong bonding effect to take place and allows the quantum well the ability to generate light as a semiconductor laser.

# Quiz

The band structure of a semiconductor is given by:

Where mc = 0.2 * m0 and mv = 0.8 * m0 and Eg = 1.6 eV. Sketch the E-k Diagram.

# de Broglie’s “Matter Waves” and Full Schroedinger’s Equation

Photons were originally assumed to be waves and even after they were found to be particles, photons still exhibited the same qualities that allowed them to be considered as waves before. de Broglie considered that, if a photon could be considered both a particle and a wave, then perhaps other forms of matter could be treated as waves.

de Broglie’s finding later became a pillar of quantum mechanics. The Schroedinger equation becomes an equation for such ‘matter waves’ as proposed by de Broglie. The plane wave as mentioned above later becomes the ‘wavefunction’ ψ(x,t)  as is fundamental to Schroedinger’s equaiton. Is the wavefunction ψ(x,t) measurable? What is the meaning of the wavefunction ψ(x,t)? To come to this understanding, we should unpack some of de Broglie’s matter wave formulas.

Consider that the wavelength is inversely proportional to the momentum in de Broglie’s formula. Further, the momentum is found to be equal to h_bar multiplied by the wavenumber.

de Broglie’s wave formula presented further complications however. It was found that certain aspects of the wave would in fact be correct, but other parts of the wave would not be accurate. For instance, the phase of de Broglie’s matter wave would be accurate. The phase of the wavelength is understood to adhere to Galilean, non-relativistic physics, meaning that the result is not altered according to perspective. An example of a system would be relitivistic is the case for instance where the speed of a vehicle is determined relative to another moving vehicle. The phase of de Broglie’s calculation is therefore is a type of ‘objective’ calculation.

Further calculations below show that the waves proposed by de Broglie are not directly measurable. The wavefunction  ψ(x,t) is also known to be multiplied by an imaginary number and this makes it difficult to measure. Further, the system is not Galilean invariant. This means that matter waves may differ according to reference. Finally it is concluded that the wavefunction ψ is not like sound waves, water waves, mechanical or electromagnetic waves. A difference according to reference however does not mean that all hope is lost in the calculation of the wavefunction, however. By taking account for the difference in reference, two points can be compared to allow for a wavefunction that may vary according to reference.

What does a matter wave look like?

Consider it should look like any other wave. It should have some kind of sinusoidal representation. Considering one aspect of matter, it should be that matter is not allowed to exist in any one place. This is the case for instance, when a particular piece of matter is not present in a location. However, to restrict a matter wave by time could be problematic. This is to say that, at a certain moment, no matter is allowed to exist anywhere or for all positions.

There are four cases listed below that, being sinusoidal, may seen to be possible respresentations for matter waves. In the case however for a sin(kx-wt) or cos(kx-wt), it is implied then that for time wt eual to pi/2, 3*pi/2, etc matter is not allowed to exist anywhere. Therefore these functions are not acceptable representations.

For the case however of e^(ikx-iwt) + e^(-kx-iwt) and it’s counterpart e(-ikx+iwt)+e^(ikx+iwt), this is not the case. Could both be used together in superposition? The answer is that if they were added together, matter would be restricted to one direction, which is undesireable as well. Therefore, either representations would be acceptable, but not both. The boxed answer below is the normative convention taken by physicists for the matter wave wavefunction for a particle. The other representation would work as well.

Principle of Stationary Phase

Consider the case of a narrow peak, modulated by a sinusoidal function centered about zero. If the sinusoid frequency is too high with relation to the narrowness of the peak or if the phase is rapidly changing, the averaging of the system will cause the narrow peak to disappear. If however the frequency is low – or if the phase is stationary at the narrow peak, any averaging that is done will still allow the narrow peak to exist.

As related to quantum mechanics, while determining characteristics of a wave, the principle of stationary phase becomes important. The wave Φ(k) as a function of wavenumber will be detected as a narrow peak. In order to properly detect this narrow peak, the modulating sinusoid must have a phase that varies much less with respect to (in this case, k) the x-axis. Otherwise, the narrow signal will be lost.

Towards developing a generalized Wavefunction

Performing a specific operation to the wavefunction interestingly produces the momentum multiplied by the wavefunction. The operator used is termed the “momentum operator.”

As we know from linear algebra, if a matrix multiplied by a vector is equal to a number multiplied by the vector, then the vector is termed an eigenvector of the matrix.

Given the above momentum operator relationship, it may be concluded that ψ(x,t) is an eignevector, or more specifically an eigenstate of the momentum operator p_hat. p is then also the eigenvalue.

And finally, using the eigenstate condition, Schroedinger’s Equation in general form for a free particle is derived:

Barton Zwiebach. 8.04 Quantum Physics I. Spring 2016. Massachusetts Institute of Technology: MIT OpenCourseWare, https://ocw.mit.edu. License: Creative Commons BY-NC-SA.

# Compton Wavelength of a Particle, Compton Scattering

Planck’s constant h is fundamental to Quantum Mechanics. But what is Planck’s constant?

Separating the fundamental units from h, it is found to be equal to a distance multiplied by a momentum. If then we consider using the speed of light, c and a given mass, we arive at a new parameter equalling a distance. This parameter is the Compton wavelength of a particle.

The Compton wavelength of light:

Compton wavelength of an electron:

Compton Scattering

Physicists had considerable difficulty accepting the existence of the photon. For one, it introduced the non-deterministic nature which collided with classical mechanics and it also went against Maxwell’s theories, which at the time were considered highly successful. The concept of Compton Scattering was one of the final pieces of support that lead to a wider acceptance of the photon as a particle.

Compton Scattering provided an image of a photon literally colliding with an electron, showing that like the electron, the photon was worthy of being considered a particle as well. The classical version of Compton Scattering was Thompson Scattering, which considered the photon as a wave. The Thompson Scattering experiment applied a photon wave to an electron, which caused the electron to be energized and radiate a photon wave.

Compton Scattering, treating the photon as a particle provides a different picture at higher energy levels:

Barton Zwiebach. 8.04 Quantum Physics I. Spring 2016. Massachusetts Institute of Technology: MIT OpenCourseWare, https://ocw.mit.edu. License: Creative Commons BY-NC-SA.

# Entanglement, Mach-Zehnder Interferometer

It may be that one wishes to describe a quantum state as the existence of two separate, non-interacting particles each in a particular state. Imagine particle 1 can be in either state |u1> or |u2> and particle 2 can be in either |v1> or |v2>. We wish to describe the state where particle 1 is in state |u1> and particle 2 is in state |u2>. The notation for this state is:

|u1> ⊗ |u2>

Given a probability for each state, a general state formula can be written to describe both particles:

To describe a superposition of particular states however will result in a dependency of the particles on each other. This is called an engtangled state.

The following outlines an entangled state example. This shows how the fate of two particles becomes intertwined in such an entangled state, where there will exist no other combination other than the complete state combinations made available by the definition of the state.

Einstein has objected to the entangled pair hypothesis. John Bell had proposed an experiment to test entanglement using a three directions, such that a correlation would be more presentable. The results of his experiments however confirmed the possibility of this sort of entanglement on the quantum level, which appears to deny classical mechanics.

Mach-Zehnder Interferometer Quantum Mechanical Calculation

Let us model the Mach-Zehnder Interferometer using photon probability state matricies. First, we will consider the operation of the beam splitter. When a photon enters the beamsplitter from one direction, there is a given probability that the photon will be present at either the transmitted or refracted position. Given that the beamsplitter is balanced, meaning that the photon has an equal change (1/2) of exiting either side, the beamsplitter is modeled below:

Next, using the beamsplitter matrix, the Mach-Zehnder Interferometer can be modeled. Interestingly, the photon appears to exit (100%) from the side opposite which it entered.

Let’s consider a case in which mirror 2 is blocked. Using the matrix for beamsplitter 1 and beamsplitter 2, the probabilities are calculated that the photon will either 1. be blocked by the concrete, 2. exit at detector 0 and 3. exit at detector 1.

Barton Zwiebach. 8.04 Quantum Physics I. Spring 2016. Massachusetts Institute of Technology: MIT OpenCourseWare, https://ocw.mit.edu. License: Creative Commons BY-NC-SA.

# Non-determinism & Superposition in Quantum Mechanics

A system is deterministic if the outcome is easily predictable. Ever since the beginning of Quantum Mechanics, precisely when Einstein had proposed the existence of photons, scientists were concerned that the quantum mechanical model of physics was not deterministic.

Einstein had proposed that light was made of quanta called photons. Previously, light had always been considered a wave (as explained by Maxwell’s equations) and now it was being considered as particles.

Einstein’s formula for the energy of a photon: E = h*ν, where ν is the frequency (ν = c/λ).

One issue arose in the case of an experiment with a polarizer. A polarizer is a system that abosrbs energy of light that is not focused in the direction of the polarizer. The question arose then, how does one determine whether a photon in a beam that forms an angle with the polarizer will go through the polarizer? In vector mechanics, only the fraction of the magnitude that is in the direction would enter. In Quantum Mechanics, presenting an idea of a fraction of a photon went against the theory. In the end, it was decided that it could not predictably be determined whether or not a particular photon would be allowed through the system and it was a system represented by probability.

The method of writing a photon moving in a particular direction is written in the following way in Quantum Mechanics:

Photon in the x direction:

• |photon; x>

Photon in the y direction:

• |photon; y>

Superposition in Quantum Mechanics

The nature of superposition in Quantum Mechanics is different than in classical mechanics.

The optical experiment using the Mach-Zehnder Interferometer bacome of interest to physicists dealing with Quantum Mechanics. Questions arose, such as how to understand the interference between photons in the interferometer. If two photons were able to interfere with each other in a cancelling manner, this would result in a violation of the conservation of energy. Likewise, a constructive interference would result in the creationof photons, which is also problematic. The current understanding that resulted from this debate was that photons interfered with themselves and that photons are unable to interfere with other photons. Further it was proposed that after moving through the beamsplitter, a single photon will exist in either path devised by the beamsplitter. As it comes to the second beamsplitter, the path of the photon towards either detector is a probabilistic expression. The existence of a single photon in both paths is the understanding of superposition in Quantum Mechanics.

In the superposition of states in the quantum mechanical model, the result of the two states is the outcome of either of the two added states with a probability between either condition. The addition of both states with a scaling factor will effect the probability of the outcome state, however there is no intermediate, average or ‘total’ outcome state. States in a quantum mechanical model are notated as |A> and |B> for states A and B.

There is an assumption in quantum mechanics that the superposition of a state on itself does not change the outcome.

Using the above assumption, we are able to alter a system of photon states in two directions to simplify the expression to one complex parameter.

Consider you would like to design a new quantum state. It is measured that two quatum states are for one system, a positive particle spin in the z direction and a negative (downward) spin in the z direction. A superposition of these two states together is referred to as a new quatum state. An experiment could be done many times to determine the probability of outcome. The spin of the particle will however only be either in the positive z direction or negative z direction.

Barton Zwiebach. 8.04 Quantum Physics I. Spring 2016. Massachusetts Institute of Technology: MIT OpenCourseWare, https://ocw.mit.edu. License: Creative Commons BY-NC-SA.

# Linearity of Quantum Mechanics, Schödinger’s Equation

Linearity

A linear function follows two properties:

1. If a function is a solution and each variable is scaled or multiplied by the same number, then this is also a solution.
2. If two solutions to a function are found, then a third solution of the function is the summation of each variable in the function.

Given a linear operator L and an unknown variable u, the following properties apply:

Here is an example:

Linearity as related to Quantum Mechanics

A linear system is far less complicated than a non-linear system.

Maxwell’s equation is linear, for instance. Newton’s equations are not linear.

Consider the example below that explains a particular scenario in which Newton’s equations are shwon to be non-linear:

Quantum Mechanics is linear. Schroedinger’s Equation, devised in 1925 for a dynamic variable, ψ termed the wavefunction.

H_hat is the Hamiltonian, a linear operator as was L in the previous example for Linearity [link]. This means that in Quantum Mechanics, solutions can easily be scaled and added together. Thus, it is proven that Quantum Mechanics is actually simpler than classical mechanics. i is the complex number operator equal to the square root of negative one and h_bar is Planck’s constant.

You might ask what the wavefunction is about, if there are any units, for instance. Interestingly, Schroedinger was not sure what the wavefunction referred to exactly. Max Born later proposed that it had to do with probability.

Complex Numbers in Quantum Physics

The complex operator i at the front of the formula notes a significant departure from classical mechanics in which almost all systems are primarily real. In the case of Quantum Mechanics, complex numbers are essential.

Euler’s formula, e^(i*x) = cos(x) + i*sin(x) also proves useful in Quantum Mechanics.

Below is some review relevant to Quantum Mechanics:

Barton Zwiebach. 8.04 Quantum Physics I. Spring 2016. Massachusetts Institute of Technology: MIT OpenCourseWare, https://ocw.mit.edu. License: Creative Commons BY-NC-SA.