Electronic Properties of Materials/Printable version


Electronic Properties of Materials

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Quantum Mechanics for Engineers

This is a section in the book Electronic Properties of Materials

Within this section there are 11 chapters planned.


Quantum Mechanics for Engineers/Quantum Mechanics Overview

This is the first chapter of the first section of the textbook Electronic Properties of Materials.


Quantum Mechanics OverviewEdit

The origins of quantum mechanics came about in the quantum revolution from 1890 to 1930. During this time several new discoveries facilitated this transition.

  1. Light has particle nature in addition to wave nature.
  2. Light (photons) and matter are found to interact and develop theory of atomic structure.
  3. Matter has wave nature in addition to particle nature.

These discoveries led to the birth of modern quantum mechanics.

Light Has A Particle NatureEdit

 

As early as 1877, Boltzman proposed that energy was not continuous, but rather discretized. In 1905, Raleigh-Jean applied this to black bodies, a perfect radiator where radiation is emitted form vibrating atoms that act as little dipoles to create the Raleigh-Jean Theory. This theory takes   as the expectation value of the energy giving:

 

This distribution is energy times the distribution over the partition function which produces  . While this generally follows experimental results at large wavelengths, at shorter wavelengths the prediction diverges from experimental results.

 

In 1901, Plank also took the existing theory and modified it to replace continuous energy with discrete energies giving:

 

This 'fix' for the UV catastrophe was completed by incorporating Wein's Law (1893).

Discrete EnergiesEdit

When we think about energies they may look continuous but they are actually discretized. Furthermore, in 1905 Einstein said that not only is energy quantized, but so is electromagnets. in 1887, Hertz showed through his photoelectric effect experiment


Quantum Mechanics for Engineers/The Stern-Gerlach Experiment

Electronic Properties of Materials
 ← Quantum Mechanics for Engineers/Quantum Mechanics Overview Printable version Quantum Mechanics for Engineers/The Fundamental Postulates → 

We discussed in the first chapter a list of historical experiments that highlight the origins of quantum mechanics. In this lecture, I want to present one final experiment. The experiment itself just showed the origin of spin and orbital quantum numbers, but we're going to have to take it a step further and discuss a thought experiment that will demonstrate the fundamental working of quantum mechanics.

The ExperimentEdit

 
Stern–Gerlach experiment: Silver atoms travelling through an inhomogeneous magnetic field, and being deflected up or down depending on their spin; (1) furnace, (2) beam of silver atoms, (3) inhomogeneous magnetic field, (4) classically expected result, (5) observed result

As it happens, for reasons we will discuss during the second half of this class, the Silver (Ag) atom has a very simple magnetic nature. Each atom can be treated as a little dipole with magnetic moment  .

<EXPLANATION OF EXPERIMENT>

The force on a magnetic moment is:

 

In the z-direction:

 

The deflection of the Ag atom is proportional to the z-component of  .

Expected ResultsEdit

Based on this, we expect to see atoms of all different orientations of  , and random magnetic moments, spread out in a single distribution.


<FIGURE> "Classic Theoretical Results of the Stern-Gerlach Experiment" (Atoms are of all different orientations of u, and there is a single distribution across the screen, centered on the main axis.

But this is not what we see...

Actual ResultsEdit

Rather, we see two separate distributions on either side of the main beam.


<FIGURE> "Actual Results of the Stern-Gerlach Experiment" (Two separate distributions, not on the main axis, are seen instead of the single, classically predicted, distribution.)

As it happens, in quantum mechanics, magnetization is tied to angular momentum. (This of electrons zipping about in a circular orbit.) In Gold we are only looking at the spin of an electron. The directional component of  , say  , can only take two values, "up"  , or "down"  . What we just did was measure   of the Silver atoms (electrons?), and separated them into two beams, one with spin-up and the other with spin-down. Is this shocking? Yes. We just took a randomly oriented vector,  , and measured it's projection,  , and found it could only take two values.

Explaining Quantum MechanicsEdit

Let's keep going. Now that (in principle) we can make a simple measurement we can make a series of thought experiments. Let's pass a beam through a filter, and see what happens...

<FIGURE> "Explaining Quantum Mechanics: The   Box" (Some beam,  , enters the box,  , and is separated based on up and down spin.)


Let's take some beam,  , have it enter the   box which separates the beam based on up and down spin. If we take the output from   measurement, discard the up elements, and remeasure down beam, the resulting beam will still be "down". This is good, no surprise here as this follows with classical logic.


<WHAT IS THIS>

Hypothesis - Polarized sunglasses all y-components are discarded.

  1. Not 50/50 in polarized light.
  2. Try rotating the box...


Now let's try rotating the   box into an   box. The   beam is still being split into up and down spin by the first  box, but now that down group is being filtered based on an   box, which is an   box that has been rotated 90°.


<FIGURE> "Explaining Quantum Mechanics: The   Component" (Note that the   box is the same as the   box, just rotated 90° to measure the y-component of the vector  .)


It looks like both boxes have a base probability of 50/50 for up or down spin. Does this make sense? Maybe?

<FIGURE> "Title" (Description)

Now we filter   to be either up or down 50/50 probability?

Something seems wrong with this picture...


Let's run one more experiment. This is the same as <FIGURE>, but now the up group coming out of the   box is again filtered through an   box. Looking at the problem, this should result in 100% down spin as the elements were tested to be 100% down spin before they entered the   box, but this is not what we see here. Instead the elements coming out of the second   box are 50/50 up and down spin.


<FIGURE> "Explaining Quantum Mechanics: The second   box." (Now the   up beam is filtered through a second   box.)

This is definitely weird.   is just some vector. If you measure the sign of  , and you can measure it again and again and again, it doesn't change. BUT after you go and measure  , if you look back at   it has once again randomized. Classically, this is like taking a bunch of marbles and splitting it into red and blue marbles. You then split the blue marbles in to large and small, but when you look back at the pile, half of the blue marbles have changed into red!

Why does this happen?Edit

The components of   are "incompatible", as we can only know one component at a time. Before we measure   we can say that the atom's wave function is in a "superposition" of being up and down. By using Born's probabilistic interpretation, or psi wave, we know that the odds of measuring up or down is 50/50. We measure   and the psi wave "collapses" to  or  , depending on the measurement. Subsequent measurements have 100% chance to repeat the initial measurement according to the probabilistic interpretation of  . In  , the system is in a superposition of being  . If we measure   and find  , then we cause the wave function to collapse to  . In this state we have no information about  . We lost the information we had measured earlier when psi collapsed into  .

In the next section we will go over the formalism of quantum mechanics, and will readdress the Stern-Gerlach experiment mathematically.


Quantum Mechanics for Engineers/The Fundamental Postulates

Electronic Properties of Materials
 ← Quantum Mechanics for Engineers/The Stern-Gerlach Experiment Printable version Quantum Mechanics for Engineers/Particle in a Box → 

There are four basic postulates that underlie quantum mechanics.

Postulate I: Observables and Operators are Related

Postulate II: Measurement collapses the Wave Function

Postulate III: There exists a state function that allows expectation values to be calculated.

Postulate IV: The wave function evolves according to the time-dependent Schrodinger equation.

Postulate IEdit

Each self-consistent, well-defined, observable has a linear operator that satisfies the eigenvalue equation,  , where   is observable,   is the operator,   is the measured eigenvalue, and   is the eigenfunction of  . In a given system you have a different eigenfunction for every eigenvalue so often times you will see   which specifies that   is the eigenfunction of  . Thus, this postulate links an observable to a mathematical operator.

What are Mathematical Operators?Edit

An "operator" is thing or mathematical expression which operates on a function and makes it different. For example:

 

In this function,   is the mathematical operator defined as the derivative with respect to  . This means that if we later have   operating on some function of  , we can then apply additional operators to the function which change the result, but still follow the same rule. For example, let's apply an operator,  , which rotates the function 90° about the z-axis.

 


Furthermore, applying a "divide by three" operator, or an Identity Operator, which leaves the function unchanged, yields similar results.

 

Physically Significant Operator Observables:Edit

Physically meaningful observables all have operators, which come about in a variety of ways, but the way that you can start to think about them is as operators in the classical world which are further quantized with the addition of   and  . If you look at these case s long enough, you'll eventually start seeing that there's a pattern to it.

Let's take the example of linear momentum,  . I will give it the operator,  , a vector which is equal to  . While you can look at the whole in three dimensions, the gradient allows us to look at it equally in parts so let's simplify this problem and look only at the x component of this vector.

 
In applying this operator to some function,  , gives:
 

Solving this differential equation provides one solution by applying the planewave equations:

 

The solution is just a planewave with wave number,  .  

 
 

This isn't very exciting on its own as   and   can take any value, thus it doesn't look "quantized". Physically, this represents a free particle (i.e. a particle alone in an infinite vacuum), and the quantization comes from the boundary conditions we apply.

Application of Boundary ConditionsEdit

<FIGURE> "Born-von Karman Boundary Conditions" (These boundary conditions could be pictured as a box or as a ring.)

Let's apply periodic boundary conditions (PBC) called "Born-von Karman Boundary Conditions". <FIGURE> With this we are essentially putting the particle in a one-dimensional box where it is free to move within the box, but once it leaves the box it loops back around in space and reenters the box from the other side. The box has some size,  , which gives us the quantization. This concept can also be pictured as a ring with radius  .

These boundary conditions restrict the solutions, because the solutions must match at these boundaries. Thus:

 
This isn't obviously solvable so we go in and substitute sine and cosine as described in the planewave equations which gives:
 
Since the right hand side of the equation must be equal to a known value, we can conclude that  . Following this logic:
 

Now we have a quantized solution. Going back to the idea of the ring boundary condition, and come upon the de Broglie hypothesis from Chapter 1 ( ), showing us that when Plank initially quantized particles he was thinking of a periodic situation. Additionally, we can develop the Bohr model of the atom by combining these two concepts.

 

<FIGURE> "Bohr Atom Model from de Broglie Equations" (Description)

Effect of Boundary ConditionsEdit

This is what makes nanoscience interesting! When the dimensions of a structure are small enough they affect the quantization. If we can control the dimensionality at a nanoscale, we can control the quantum nature of electrons.

Another well defined observable is energy. In classical mechanics there are several ways to formulate the equations of motion (Newtonian, Lagrangian, Hamiltonian). I'm not going to talk about these, but you should know that in quantum mechanics the formalism matches classical Hamiltonian formalism. For systems where the kinetic energy depends on momentum and potential energy or position, the Hamiltonian operator takes the simple form:

 , where   is the kinetic energy and   is the potential energy.

For now we are going to talk about particles in a vacuum which sets the potential energy ( ) to zero. For now we are simply looking at the kinetic energy ( ). We can take the equation for kinetic energy,  , from classical mechanics and substitute in our momentum operator,  , to get a simplified equation for  , referred to as the Laplacian operator.

 

Simplification of nabla^2:

 
Once again, we can simplify this to a one dimensional problem, by utilizing the expanded form of  .
 

We are taking the second derivatives so as the operator is operating it returns the curvature of the function showing us that the kinetic energy operator is proportional to a function's curvature. Thus, solutions with tighter curves will have higher energies than slowly changing functions.

Ideally, we want to solve:   (Time-Independent Schrodinger Equation)

 

What solves this? Planewaves!   As it turns out, planewaves are a common solution in quantum mechanics!

 


Here we can see that our eigenvalues are  , thus breaking up the equation gives us:

 

These variables are consistent with our earlier finding that:

 
Note: Our earlier equation  had one component due to the singe derivative present in the parent equation while our current solution has two components due to the double derivative present in the parent equation.

Here, the momentum is telling us what the value is and the   and   coefficients are telling us if it travels to the left or to the right. As you may have guessed, the energy and the momentum are commensurate with each other, we can know them both at the same time. In quantum mechanics, if operators "commute" then they share eigenfunctions. We should notice that if   or   are zero, then the eigenfunctions of energy are also the eigenfunctions of momentum. Generally,   and   commute if:

 
For example, let's look at momentum and energy, when   is some test function:
 

Since  ,   and   commute.

Let's try a different operator. This time, let's compare position and momentum.

 

Here,  , meaning that   and   do not commute. This means that momentum and position do not commute and thus do not share eigenfunctions. As it so happens, this is all tied to observation and the fundamental uncertainty in our knowledge.

Recall the Heisenberg Uncertainty Principle:

 
When operators commute then we say that the observables associated with the operators are "compatible" meaning that they can be measured simultaneously to arbitrary precision. (Related to the Schwartz inequality...) Without proof, I will tell you that:

If  , then  , where   refers to "expectation value".

So, for  ,   (working with  ) *see B&J p.215

This is a BIG DEAL! It means that it is impossible to simultaneously know certain things. (Remember our thought experiment from Chapter 2?) What's more, this is purely a quantum effect. Consider again, momentum. What if we precisely measure the momentum to be  , then the particle's wave function is  .

Remember in the probabilistic interpretation:

 
<FIGURE> "Incompatible Observables" (Constant value  )

But   is just the normalization constant, so the probability distribution appears as (FIGURE). If we know precisely   then we know nothing about  ! It was an equal probability any where in the range  .

Thus,   and   are incompatible observables.

Postulate IIEdit

A measurement of observable   that yields value   leaves the system in state  .

 

We say that the measurement "collapses the wave function" to  , where   is the eigenfunction of the particular value measured Immediate subsequent measurements will thus yield the value   as the eigenfunction will remain collapsed about that value   until another property is measured, as seen in Chapter 2.

What is important here? Before the initial measurement, the expectation of the measurement is given statistically from  , a superposition of possible states. After the act of measuring leaves  , one particular state, for subsequent measurements. Note that this is very similar to solving partial differential equations. When solving a partial differential equation for a particular solution you get a linear superposition of all possible solutions which is analogues to what we see here.

Postulate IIIEdit

There exists a state function, called the "wave function" that represents the state of the system at any given instant, and all the information we could know about the system is contained in this state function,  , which is continuous and differentiable.

For any observable,  , we can find the expectation value, for measuring   from  .

 
Here   is the complex conjugate of  , and   is an abbreviation for  

Review of Statistics (and the meaning of the "expectation value",  )Edit

In statistics,  , is the expectation value of,  , and when all goes well in sampling theory:

 

Within this function, if you know all the possibilities then you can essentially write the state function for the system. Let's say I have a bag with 5 pennies, 3 dimes, and 2 quarters. The probability of me pulling any given coin type out of the bag is:

 
 

For continuous probability distribution:

 

State Functions in Quantum MechanicsEdit

Applying this statistical expectation value to our quantum state function gives us:

 

Where, since   is just a number we can simplify   to  .

Postulate IVEdit

The state function,  , develops according to the equation:

 

This is the time dependent Schrodinger Equation and is true for non-relativistic space. (Note that this equation is a postulate, there is no proof for this.) As it happens, to account for relativity we either fix our solutions by perturbation methods or instead solve using the Dirac Equation:

 

These four postulates give us the basis for everything we do in Quantum Mechanics, and the reason they work out is tied to linear Hermitian operators. The solution to the eigenvalue equation has special properties, wherein the eigenfunctions are orthonormal. For an arbitrary system with bound states:

 ; where  , and   is the   eigenvalue which corresponds to the   eigenfunction  .

OrthonormalityEdit

An orthonormal function...

 


Here,  , is the Kronecker Delta Function. This function is a consequence of the Stern-Louisville Theorem where the set of ident functions,  , span Hilbert space, sometimes only sub-space, the function-space where   lives. Hilbert space can be thought of as an equivalent space to Euclidean space, where vectors live, which will have some set of vectors  . If that set of vectors is orthonormal and span space, then they can act as a basis for all other vectors in that space, and we can write any arbitrary vector   as a sum of these vectors  .

 

Those who have taken linear algebra might also remember a bunch of rules about eigenvalues, pertinents, etc... Well, they all will apply to what you're going to see here, and in fact, there is a matrix notation that allows one to directly map all of quantum mechanics to sets of matrices and vectors.

Hilbert SpaceEdit

With this orthogonal property, we can express   using   as a basis.

 

Just as with Euclidean space,   are the projection of   onto  . The value of this being that we can solve for   by taking the equivalent of an inner product. (dot product)

 

The fact that we can have a basis which is orthonormal, spans space, allows us to write the wave function, gives us a way to describe it in Hilbert space, and allows us to describe the coefficients as the projection of the wave function onto that particular eigenfunction, is very important!

Think back to expectation values, where  . Solving for each term:

 
Thus,  

Therefore the probability of measuring a particular value is  , given by the coefficient which is the projection of the wave function onto that particular eigenfunction. If you think about this physically in vector space, it kind of makes sense! We're saying that if I have a vector that's mostly in the 1 direction, then it's going to have a behavior that's also "mostly" in the 1 direction. There is still a probability of measuring it in the other directions as well. So, when we talk about superposition, it's as a linear sum of eigenfunctions. Remembering that with each eigenfunction there is a coefficient which is the projection of the wave function onto that eigenfunction, this tells us the probability of measuring any particular value.

Return to Stern-GerlachEdit

We have some operator,  , which operates on some function,  , and returns the value  . This system has only two solutions (in the case of the silver atom):

 

When we had that initial beam of atoms, passing through vacuum, initially we didn't know anything about the state; it was randomized.

 

This says that the probability of measuring each outcome is 50/50 odds! Furthermore, the wave function is normalized and the sum of the probabilities is equal to one. If this was not true we would have to go through and scale the vector until it is normalized. Now let's say we measure the case and find an "up" spin, meaning that   has collapsed to  . Now that we have measured the case, the probability of further finding an "up" case is now one and the probability of finding a down case is now zero.

What about  ?

 ;  

This system has two possible results, analogous to the ones shown with  . We can write both systems together as:

 

The set   and   are incompatible. When we measure one, the vector function snaps to one of the basis, then again with the other.

Most importantly, we can collapse   into either   or  , but not both. These two operators are incommensurate, as they don't commute, and if they don't commute they must form different basis sets within Hilbert space. We can write them out sideways, as each set is still equal to the wave function, but information about one set does not tell us anything about the other set.

The collapse of   to   or   is unique to quantum mechanics and is why we can't simultaneously know these two observables!


Quantum Mechanics for Engineers/Particle in a Box

This is the fourth chapter of the first section of the book Electronic Properties of Materials.

<ROUGH DRAFT>

So far we've gotten a feel for how the quantum world works and we've walked through the mathematical formalism, but for a theory to be any good, it must be possible to calculate meaningful values. The goal of this course is to show how the properties of solids come from quantum mechanics and the properties of atoms. Before we look at the properties of solids, we need to study how electrons and atoms interact in a quantum picture. Over the next several chapters, we will study this, but first we need to consider atoms in isolation.

So we want to solve the time-independent Schrodinger Equation,  . As it happens, finding   for most problems is non-trivial. In atoms, the FIGURE potential goes  , but the FIGURE interactions are difficult, as we will prove later. As it happens, the way to approach this is through simplifications and approximations. We're going to start with the simplest calculations and build up from there.

Time-Dependent Schrodinger EquationEdit

 
Particle in a 1D Box

Let's look at a particle in a one-dimensional box with infinite boundaries.

 

The fact that   only goes from zero to infinity means that we can essentially throw out anything past the defined barriers of our box. Note that here we will solve for   only, not  , which implies a separation of variables. Let's check this idea by guessing the solution:

 

Here the solution is a product of two functions,   and  . To solve, we substitute it into the time-dependent S.E. and rearrange.

 


Both pure t,  , and pure x,  , must be equal to some shared constant,  .

Thus:

 


< ???>

Look! It's the Time-Independent Schrodinger Equation! This is exactly what we want to solve. As the Hamiltonian operator is the operator of energy we're going to be getting eigenvalues,  , which are measurable values of energy, and eigenfunctions,  , which are the functions corresponding to the energy. It is common for people to rewrite this as:

 


Returning to the time-dependent part, and rewriting as:

 


<CHECK MATH - VIDEO 13:10>

Taking   as our guess, one solution is:

 

Always, when  , a solution is:

 

The Time-Independent Solution,  .Edit

The general method to solve this type of problem is to break the space into parts with boundary conditions; each region having its own solution. Then, since the boundaries are what give us the quantization, we use the region interfaces to solve.

<FIGURE> "Title" (Description)

Equations
   
   

 

<PICK ONE^^^>


Regions I and III have a fairly simple solution here:

 

Region II has:

 

What is a good solution? Let's try planewaves! The general Solution for Planewaves,  , is not very easy to lug around, and wave functions in quantum mechanics are in general complex form.

 

Now apply some boundary conditions...


 

 

 

 

Thus we have the equation for quantized energy, where n is limited to counting numbers ( ), but we still need to solve for  . Given:

 

Pick a constant to fix normalization. In this case we choose  .

 

Substitute and solve...

 


At the end of the day, we have:

 

Complex NumbersEdit

As a point of honesty, while this solution is true, there are other solutions. Not only can you just put in different values for  , but we can also change the phase of our solution. In quantum mechanics, you will often year that you're solving something to "within the factor of the phase," and when we say about that, we're talking about the phase within complex number space.   is a complex number, be we don't pay attention to the phase of the number. In other words, we can add an arbitrary   in front of   without consequence.

 

<Phi* Phi vs Phi^2>

Why? Because we can only measure the magnitude of   as  . However, in certain situations where we are comparing two  , we can measure the difference in their phase. In this course, and most of the time, we just ignore the arbitrary phase factor,  , and say that we know   to within an arbitrary phase factor.

So now we have a solution,  , but the Schrodinger Equation is a linear PDE. What does this mean? If   and   are both solutions to a linear PDE, then  . Also, in our case we have an infinite number of solutions since  , really we need to say that the general solution is:

 , were   is our solution and   are coefficients.

In addition the solutions orthogonal to one another, which is yet another property of linear PDE. This means that:

 , where   is another Kronecker Delta function.

The orthogonality of the eigenfunctions is physically important, and mathematically useful, as will be seen.

Finding the CoefficientsEdit

Returning to the problem at hand, how do we determine the coefficients  ? By solving as an initial value problem. Say that at time   we make some measurement that gives us   then project the   onto the individual eigenfunctions. So...

 


Where   is the eigenfunction of energy. Now we take:

 

So for each  , one can find   by integrating   using orthogonality of  .


What if I measure the energy? The wave function collapses to an eigenfunction of energy.

What does this mean? We can only measure quantized values. ( )

If I measure  , then

 

  (The Probability Distribution of Position)

<FIGURE> "Title" (Description)

Where is the particle? Somewhere given by the   equation. Remember  , and  , do not commute.

If I measured   instead of  , I would find a distribution of  . What is the value of energy after measuring  ? We don't know! A measurement of   causes us to lose our knowledge of  . When   is written as a summation of multiple eigenfunctions we say that   is a "superposition" of states. We don't know which state it is in, but we know it has a probability of being in one of the states in the expansion.

Imagine we know that the system is in a state:

 , where   are eigenfunction of energy.

What is the expectation of energy? Remember that  .

 
Simplifying each term:
 

But remember, we also talk about expectation values:

 

So...

 

This means that if we know  , we can determine the probability to measure any   by projecting   onto eigenfunctions of  , and  . When we have uncertainty, for example if we don't know if it's energy state one or energy state three, we have a superposition which is saying that we're taking a sum of eigenvalues.

An interesting experiment is to input this problem into Excel, python or any number or computational programmers, and make the given well smaller and smaller. As the well gets smaller, the energies will diverge and the sum becomes absolutely huge. Conversely, as the well gets wider, you will see a convergence to a value at a relatively small sum. You loose information about energy as you increase confinement.

<gif?^^^>

A Note on Hilbert SpaceEdit

The way I'm talking about   and   sounds very much like some vector-type language. In truth,   lives in Hilbert space. This is an infinite dimensional function space where each direction is some function  , and we can talk about representing   as a linear sum of   with the coeppilien for each   being the projection of   on  .

<FIGURE> "Title" (Description)

Then in Hilbert space   must be the, dot friendly, inner product that gives the projection. Measuring must move   to lie directly on  . There are other, incompatible functions,  , in Hilbert space such that both  , and   are complete orthogonal sets, and I can express   in terms of either.

 


Measuring   means losing information about   but projecting   onto on of the   directly, and measuring   looses information about  .


Quantum Mechanics for Engineers/Momentum Velocity and Position

Electronic Properties of Materials
 ← Quantum Mechanics for Engineers/Particle in a Box Printable version Quantum Mechanics for Engineers/Degeneracy → 


** ROUGH DRAFT**

Next, we are going to talk about momentum, position. What makes this discussion particularly useful is that it provides the basis for later parts of this course where we start talking about the velocity of electrons moving in a material; relevant to the conductivity of a material. First we must define velocity in quantum mechanics in terms of position and momentum.

A Closer Look at the Free ParticleEdit

Looking back at our free particle from <CHAPTER>. We solved the free particle already and our resulting Hamiltonian was  . (Note that this is for a 1D particle. The solutions are valid for 2D and 3D, but for the purpose of this exercise we will constrain ourselves to 1D.)

Additionally, our wave function,   where   gives the time evolution of the system, is separable. You can prove to yourselves that substituting these in will definitely get a Schrodinger equation that separates into a time dependent and position dependent part. Furthermore, our time dependent part looks like:  . Here   must be dimensionless, which means that   is the frequency ( ) with units of  , and  .

Similarly in blackbody radiation, we use  , or  , where   is the reduced Plank constant. By multiplying this by the relationship between frequency and angular frequency,  , we get  .


Going back to the position dependent part of our original equation, we know that this is just another planewave, as proved in <CHAPTER>. Once again our planewave function was:  , and our solution was:  . In this case, because it is a free particle,   is a continuous variable; we haven't quantized this at all.


<MATH CHECK>

We know also that momentum and energy commute:  

In fact we solved momentum in 1D which provided us the solution  , where   is still a continuum value. In this case it could be positive infinity or minus infinity. Just remember that   in the case of a planewave is the wave vector, and it's telling you the direction and the wavelength of the wave.

Finally, we also found that  , for the free particle, which means that if we measure a particular value of momentum, for instance, we will get a particular value for   ( ). Once we measure this particular value, the wave function collapses and we can write the solution as:

 
Having the commutation of energy and momentum equal zero means that we can simultaneously measure these two properties.

<MATH CHECK^^^>

VelocityEdit

<FIGURE> "Classic Particle Movement" (Description)

So let's say we've got a particular value we'll call  , and this free particle is going to have some sinusoid,  . <FIGURE> If we wait some small amount of time, and look at it again, the wave will have propagated. This is, after all, what planewaves do. Now let's say that after that "certain amount of time" the planewave propagated by some  , shown in <FIGURE>, which means that this new sinusoid is now  , where  . Essentially, if there is some  , it can be rewritten as  .

<MATH CHECK> sin wave propagation (+) or (-)

<FIGURE> "Sinewave Translation" (Propagates towards +x)

Now, if this wave is propagating, then we can talk about the velocity which also propagates. From wave mechanics, we have that the velocity is equal to the angular frequency divided by the wave vector ( ). Multiplying both the top and the bottom by  , and substituting variables from our eigenfunction solution gives us:

 

<ASIDE> Extra math from notes with no home:

 
<END ASIDE>

Particles vs. PlanewavesEdit

In this instance we solved for a delocalized particle, and found the phase velocity. Notice how this equation describes the relationship between the classical velocity ( ) of a particle and the velocity of the propagation of a particular planewave, referred to as the phase velocity ( ). Most importantly, these two velocities are NOT the same. As it turns out, a real particle will be localized.

 
Schrödinger equation wave packet

When talking about the particles that we are interested in, which have a classical velocity, they simply don't travel as a particle, they travel as a wave packet. <FIGURE> Inside these wave packets there are lots of waves with different   values, and the packet as a while moves with the same group velocity   equivalent to our classical velocity.

Particles are not single planewaves. They are a superposition of planewaves, and tend to group themselves together in these wave packets which have a group velocity of the entire group of waves in superposition. Additionally, they are within some sort of envelope function which also travels at the group velocity, equivalent to the classical velocity.

SuperpositionEdit

<MATH CHECK> phi or psi in linear wave state equation?

Imagine a superposition of plane waves. In our first example the states in superposition were discrete. They were a summation of states where  . This form has our wave function as a linear superposition of wave states ( ). Each state is a particular solution to our Schrödinger equation where each coefficient provides us with a projection of the wavefunction into these particular basis states. (Thinking back to our eigenfunctions as a basis in Hilbert space.)

This equation is equal to the infinite sum:  . Note that most of the time, when dealing in practical matters, energy is considered finite so infinite distributions are rare.

<VOCAB> dispertized?

Alternatively, instead of thinking about energy, which is <dispertized>, we generally talk about a continuous distribution. For example, if instead of talking about energies, we express this in terms of momentum. As we saw already, a particle in free space can take any value for momentum giving us the continuous distribution:

 

Here we simply replaced the sum from the infinite energy equation with an integral and integrated over all the allowed values of momentum. The resulting equation is that of the wave packet. Here   is our coefficient. This is a direct analogy to the summation from earlier as now instead of summing over all these coefficients, we are integrating over them instead, but what are these coefficients?

<FIGURE>

This coefficient is simply a function of  , representing the probabilities of finding the particle in one of these particular states. It can be thought of as  . Physically this describes the distribution shown in <FIGURE> where the probability of measuring the particle at a particular momentum is related to the value of our coefficient in front of our basis states.

Now let's simplify our equation and say that  , providing us with:

 

Looking at this solution, we know that the whole wave function, and the coefficients  , must be well-behaved. The coefficients are well-behaved as they are just some statistical distribution, going out to zero on either end and integrating to one. On the other hand,   will oscillate rapidly, so the only way that our wave function can be well-behaved on the whole is if   is a constant defined by:

 
Solving this relationship looks like:
 

Looking simply at the units in the final equation, we have  , meaning that   has the units   or   ( ). Going back to our definitions of energy and momentum we can further transform  :

 

Here,   and   are called "dispersion relations". They are essentially the energy/velocity of a particle vs. the wave number,  . They are important and researchers spend huge amounts of time, money, and resources to determine them for various material systems. For example, the band structure of a material is a dispersion relation. <CHAPTER REF> The group velocity,  , is the scope of the dispersion. When we talk about electrons moving in a crystal we talk about the group velocity, the magnitude of which generally depends on  .

<FIGURE> "Title" (Description)

The Momentum Space RepresentationEdit

Looking closer at these wave packets, let's begin by rewriting our planewave equation, putting time dependence into the general coefficient function and setting   to get rid of the energy variable. This results in the   equation:

 

Now let's apply a Fourier Transform to our equation:  


Putting this transformation into the above planewave equation results in:

 

If the set   are orthogonal to one another and normalized, then   are   also. We refer to this as the momentum-space representation of the wavefunction and Fourier space has certain properties which makes this representation extremely useful. Truthfully, there is only one wavefunction, (it is a state function!!) but here it is projected on to momentum representation where as   is projected onto position representation.

Let's consider a physically meaningful distribution. In this case, the equation for gaussian momentum is:

 

<FIGURE> "Gaussian Momentum" (Description)

<MATH CHECK> Everything below...

To define  , let's use a well-known relationship:  

 

Using a "well-known" relationship to find  :


 

 

thus...

 

Substituting this into   and solving...

 

  is itself a gaussian centered on  .

Width of a Gaussian:  

 

As   becomes large,   becomes small and vice versa.

In the limit,  

 


Watch the evolution of the   over time...

Substitute   into  

If you remember,   is our plane wave solution from <LINK>.

Solving the integral gives us:

 

Plot  

Just because you're a theorist doesn't mean you shouldn't learn by experimentation. Let's put some numbers in and see how this wave function behaves.

<FIGURE> "Example Graph 1" (t=0)

<FIGURE> "Example Graph 2" (t=5000)

<FIGURE> "Example Graph 3" (t=10000)


Quantum Mechanics for Engineers/Degeneracy

Degeneracy is often talked about in electronics and quantum mechanic in reference to electrons which have the same energy level. In this case, since energy is an eigenvalue, you end up with two electrons with different eigenfunctions which still share the same eigenvalue. We call a quantum state "degenerate" if two or more eigenfunctions have the same eigenvalue as in the case of the electrons, but how does this happen? Well, there are three separate ways; symmetry, exchange, and accidental.

Degeneracy by SymmetryEdit

This is the form of degeneracy associated with the hybridization of orbitals. Atoms behavior in the x, y and z-directions is the same assuming a spherical potential which generally applies to atoms in isolation.

<FIGURE> "Particle in a 2D Box" (Description)

Imagine a particle in another box, once again with infinite potential on all sides, but this time, in a 2D, giving us the Hamiltonian equation:

 

This problem easily breaks into component parts:  

Substituting in the Schrödinger equation and working through the math finds that:

 

Looking at these time-independent eigenfunctions, when  , then we find that   but