Quantum Mechanics and its Applications
Px262: Quantum Mechanics and its Applications
Written by Lewis Baker September 2011
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Quantum Mechanics and its Applications
Contents Introduction ..................................................................... .......................................................................................................................................... ....................................................................... .. 4 One dimensional Schrödinger equation ............................................................................................. ............................................................................................. 4 Infinite potential well ......................................................................... .......................................................................................................................... ................................................. 4 Finite potential well ............................................................... ............................................................................................................................ ............................................................. 6 Harmonic Oscillator ............................................................... ............................................................................................................................ ............................................................. 6 Three dimensional Schrödinger equation................................................................... ........................................................................................... ........................ 7 Spherical polar stuff ............................................................................................................... ............................................................................................................................ ............. 7 Quantum numbers ........................................................................................ .............................................................................................................................. ...................................... 7 Back to spherical stuff ............................................................ ......................................................................................................................... ............................................................. 8 Postulates of Quantum Mechanics ............................................................................. ..................................................................................................... ........................ 8 Postulate 1 .............................................................. ..................................................................................................................................... ......................................................................... 8 Postulate 2 .............................................................. ..................................................................................................................................... ......................................................................... 9 Postulate 3 .............................................................. ..................................................................................................................................... ......................................................................... 9 Postulate 4 .............................................................. ..................................................................................................................................... ......................................................................... 9 Postulate 5 .............................................................. ..................................................................................................................................... ......................................................................... 9 Dynamical Variables .................................................................................................... ............................................................................................................................ ........................ 9 *Expectation values .................................................................................................... .......................................................................................................................... ...................... 10 Commutation relations ..................................................................................................................... ..................................................................................................................... 11 Angular momentum ............................................................................................................... .......................................................................................................................... ........... 11 Ladder operators ............................................................................ ........................................................................................................................... ............................................... 12 Matrix representation ................................................................................... ....................................................................................................................... .................................... 13 Zeeman Effect ............................................................................................................. ................................................................................................................................... ...................... 14 Spin Orbit Coupling/ Anomalous Zeeman Effect .............................................................................. 15 Proofs .................................................................... ......................................................................................................................................... ................................................................................ ........... 15 Proof that eigenvalues are orthogonal ............................................................................................. ............................................................................................. 15 Proof that 2 observables have common eigenfunctions and they commute .................................. 17
Proof that [Lx, Ly] = i Lz. ..................................................................................................................... ..................................................................................................................... 18 Is an operator Hermitian? .................................................................. ................................................................................................................. ............................................... 19 Expectation values ............................................................................................................................ ............................................................................................................................ 20 Density of states survival guide! ............................................................................................ ....................................................................................................... ........... 21 Applications of Quantum Mechanics ................................................................... .................................................................................................... ................................. 22 Introduction ................................................................. ...................................................................................................................................... ..................................................................... 22 Pauli Exclusion Principle and fermions ......................................................................................... 22 Page 2 of 28
Quantum Mechanics and its Applications
Metals and Free Fermions ................................................................. ................................................................................................................ ............................................... 22 Fermi Wave W ave Vector ............................................................ ....................................................................................................................... ........................................................... 22 Density of states........................................... states................................................................................................................. ................................................................................. ........... 23 Conserved densities .......................................................................................................................... .......................................................................................................................... 24 Magnetic Susceptibility ..................................................................................................................... ..................................................................................................................... 24 The Nucleus and Binding Energy...................................................................... Energy....................................................................................................... ................................. 24 Insulators and Semiconductors............................................... Semiconductors......................................................................................................... .......................................................... 25 Standard Model ...................................................................... ................................................................................................................................ .......................................................... 26 Klein Gordon Equation ......................................................................................... .................................................................................................................. ......................... 27 Dirac equation ........................................................................................... ............................................................................................................................... .................................... 27 The nuclear force .............................................................................................................................. .............................................................................................................................. 28
Disclaimer: This guide has been constructed to aid revision for the module in question. All content has been checked briefly but I cannot guarantee it is all accurate. Hopefully you will find it useful and a good place to start or to compliment your revision. Remember, using your own notes and other sources will help you to revise and will reduce the likelihood of copying any mistakes in this guide. Good luck!
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Quantum Mechanics and its Applications
Introduction Quantum is a bit nuts. I would like to end it there but it probably won’t help much with the exams, although it is a perfectly good statement. This guide is split into three main sections, fo rmal quantum mechanics, Proofs and the applications of quantum mechanics. The module is quite big and I feel it is most important to provide more of a summary of it all rather than in-depth derivation on all aspects!
Formal Quantum mechanics One dimensional Schrödinger equation From last year’s quantum module, you should have met the 1 -D time dependant Schrodinger equation (or simply the 1-D Shr equation), given by:
Here, i , is the imaginary number, Ψ, is the wave function, m is the mass of the particle (usually an electron) and V, is some kind of potential for example, a harmonic potential well. Solutions to this equation are possible wave functions for the system in question. For example, a solution exists of the form:
There are a couple of things which have to be noted about a wave function. A wave function must satisfy the following conditions:
The wave function must be continuous, single valued function of position and time.
The integral of the squared modulus of the wave function over all values of x, must be finite this leads to the normalisation condition:
The first derivative of the wave function must be continuous with respect to x (unless there is a infinite discontinuity in the potential).
The second point of this is particularly note worthy. It basically is saying, there is a 100% chance of finding the particle within all space (sounds obvious but it is important as a normalisation function which is always true).
So let’s put this all to use and actually solve something!...
Infinite potential well
This is a nice problem to solve, so we consider a potential well (see the diagram below) of the form,
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Quantum Mechanics and its Applications Ok so for a infinite potential well, V = 0 inside the well and V = ∞
V
outside the well. Well for a start, the wave function must be zero outside the well! This is obvious from the fact V = ∞.
So what about inside the well, where V = 0. Take the 1-D shr equation (1). It reduces because the system has no time dependence. Therefore we arrive at the time independent Schrödinger equation:
Where, E , is the total energy of the system.
Since V = 0 inside the well it reduces further to:
-a
a
Ok so now let’s rearrange equation (7):
Notice this is just a nice simple second order differential equation which can be solved as follows:
Where...
We have two boundary conditions, for X = a and X = -a our wave function, Ψ = 0 (this is a good thing). So plugging these boundary conditions into equation (9) we find that...
(note that for (12) i used that Acos(-Ka) = Acos(Ka) and Bsin(-Ka) = -Bsin(Ka)) So now we can see that if A=0 then sin(Ka) = 0 so Kn = nπ/2a for n= 2,4,6... and if B=0 then cos(Ka)=0 and so K n = nπ/2a for n = 1,3,5... From equation (10) and now knowing that K = n π/2a we arrive at our awesome, QUANTISED answer:
That is it for the infinite potential well, now how about the finite potential well... Page 5 of 28
Quantum Mechanics and its Applications
Finite potential well So here we have some potential well which isn’t quite as simple but we go about solving it much the same as before. Say I have a potential well as follows:
So for x
-a we have like before:
But for outside the boundary i.e.
we have a slightly different Schrödinger equation:
Then just a bit of rearranging to arrive at:
The solution is then of the form of:
I won’t bother to completely solve it further than this as frankly the physics can now be seen, the solution is decaying exponentially which is what is expected. This shows that the wave function outside the potential decays exponentially, woo.
Harmonic Oscillator This is simply a description a particle moving in a harmonic potential for example a potential in the form of:
Notice the similarity to hooke’s equation :). Ok so now the Schrödinger equation for this would be:
Solving this we find:
This shows that the energy in the harmonic oscillator is quantised in units of This leads to the concept of zero point energy E 0.
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Quantum Mechanics and its Applications
It is also important to note that for large n’s the system behaves classically.
Three dimensional Schrödinger equation The 3 dimensional time dependant Schrödinger equation is:
In 3 dimensional space with volume element dV:
And like with equation (3) we have a normalisation equation for 3-D space:
We would then proceed with the separation of variables technique to solve equation (22).
Spherical polar stuff The conversion between Cartesian and spherical polar c o-ordinates is as follows:
The 3-D time independent Schrödinger equation (22) becomes in polar co-ordinates:
i nternet) we If we go ahead and solve this (I won’t do that but there are plenty of derivations on the internet) end up with 3 solutions each of which has a boundary condition which leads to the quantum numbers. These quantum numbers are incredibly useful and important and not too hard to learn either!
Quantum numbers There are 4 quantum numbers which we use in formal quantum mechanics. Each has a corresponding boundary condition which is very important to remember. The first quantum number is the principle denoted by n. It’s boundary condition is that it can take the form of any integer. 4
The second is orbital angular momentum, denoted by l . It’s boundary condition is that it takes any integer which is between 0 and n-1 (that is one integer less than the principle quantum number of the system in question).
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Quantum Mechanics and its Applications
The third is called the magnetic angular momentum, denoted by mL. It’s boundary condition is that it can be any integer which is between 0 and ± l (that (that is plus or minus the orbital quantum number).
The fourth and final quantum number is spin, denoted by ms. It’s boundary condition is that it can only take the values of plus ½ or minus ½ .
Back to spherical stuff
As I was saying, we obtain these solutions to the Schrödinger equation. These couple with the quantum numbers lead to a set of expressions called Spherical Harmonics which take the form of the following:
Where, L and m L are the orbital and magnetic quantum numbers discussed previously. These harmonics can be normalised using the following integral, it follows the same as the normalisation of the wave function, and we are saying we can find the particle somewhere is all space.
This is a important step to do, especially in an exam if you are given or asked to calculate a wave function or a harmonic and they DO NOT specific i t is normalised. If you are not told it is normalised, you should assume it isn’t. (I forgot to do this in my exam :$ ... whoops).
The normalised spherical harmonics give us the shape of the electron orbital’s we learnt in chemistry or physics A-level, how cool!
Postulates of Quantum Mechanics There are 5 postulates of Quantum Mechanics, you normally only really need to know postulates 1,2 and 4 but I will note all 5 for completeness. Postulate 1
“For every dynamical system there exists a wave function that is a continuous, square integrable, single valued function of the parameters of the system and time, from which all information of the system is contained.” This is fairly self explanatory. It states that a wave function is differentiable and contains all information about the system.
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Quantum Mechanics and its Applications
Postulate 2
“Every dynamical variable may be represented by a Hermitian operator whose eigenvalues represent the possible results of carrying out a measurement of the value of the dynamical variable. Immediately after such a measurement the wave function of the system is identical to the
eigenfunction corresponding corresponding to the eigenvalue obtained.” I like to imagine this postulate in a slightly different way. A wave function can be described as a sum of all possible states i.e.
So say an electron could be behind the TV or on top of your laptop or inside this pdf document. then n = 3 ,when we take a measurement of where the electron is we force the wave function into one of those states hence it must be in ONE not all of those states and this is the eigenvalue measured. This is effectively what the postulate is saying :).
Postulate 3
“The operators representing position and momentum of a particle are, r , and – i respectively. Operators representing other dynamical quantities bear the same functional relation to these as do the corresponding classical quantities to classical position and momentum variables.” This is saying that the operators which work in the quantum world must also work in the same way as do the classical description. This is a way of stating Bohr’s correspondence principle. Postulate 4
“When a measurement of a dynamical variable represented by a Hermitian operator is carried out on a system whose wave function is Ψ, then the probability of the result which has an eigenvalue, q n
will be the square modulus of that eigenvalue, hence, eigenfunction of the operator
. Where
and
is the
to the corresponding eigenvalue qn.”
So like equation (34) is saying, the probability of some state is equal to the modulus of the coefficient of a particular state, squared. Postulate 5
“Between measurements, measurements, the development of the wave function with time is governed by the time dependant Schrödinger equation.”
Dynamical Variables This particular topic resolves around an important relation, this is the eigenvalue equation:
is the operator, Ψ is the wave function, En is the eigenvalue.
The idea is that an operators acts on a wave function, to yield a eigenvalue of the system for example, a momentum operators will yield the value of the momentum (the eigenvalue). The idea is that we can have dynamical variables to calculate properties of a system for example,
linear momentum.
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But by postulate 3, the kinetic energy must be equal to the quantum form hence:
Hence by equating and rearranging (36) and (37):
We can do this for all our variables. But our operators must be Hermitian if they are to represent real values to be measured (postulate 2) A Hermitian function satisfies the following conditions: Has real eigenvalues Has orthogonal eigenfunctions And finally an important relation:
I note here that * denotes a complex conjugate. I mentioned orthogonal eigenfunctions, if this is to be true the following is satisfied:
Where there is an overlap function given by:
Expectation values
This is the average value we would expect to obtain from many measurements of a system when an operator
is acting on it hence the following can be shown:
*
So we re-write each wave function, Ψ and Ψ in terms on the sum of their eigenfunctions, this is what equation (34) says. Equation (34) also is usually the first step in many of the proofs in this module! To finish off this bit of proof we then proceed as follows:
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Quantum Mechanics and its Applications
Therefore we have shown that..
Commutation relations
Commuter relations are important in quantum mechanics. We calculate them quite easily like this: The commutator of two operators
is calculated by:
There is a very important consequence we find when we work out the value of a commutator.
If the value of a the commutator is zero the measurement is compatible
If the value of the commutator is non-zero the measurement is incompatible!
What does this mean? Well if a measurement is incompatible then the two bit of information you are trying to measure cannot be measured at the same time, for example, position and momentum of an electron. The leads to the idea that if two physical observables are said to be compatible, they have a common set of eigenfunctions and the uncertainty in given measurements in quantified by the following relationship: For two measurements, q and r there is an associated uncertainty of:
It is worth remembering a couple of important relations these are...
Equation (47) shows that we cannot know both the position and momentum of a particle at the same time whilst equation (48) shows that we can know two components of position OR momentum.
Angular momentum For quantum systems there are two important equations constantly used. Total angular momentum is given by where l is the orbital quantum number:
The second is the Z component of angular momentum where m l is the magnetic quantum number:
This is sometimes useful to display the possible values a system can be at, graphically.
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Quantum Mechanics and its Applications
Let’s say that that I have an electron in a state described by the quantum number n=3. From our knowledge of quantum numbers (hint equation (29)), we know that t he possible orbital quantum numbers can be n-1 hence can be 0,1 or 2 and the magnetic quantum numbers can be ± l hence hence -2,1,0,1 and 2. So the possible values of L and L Z are as follows:
For l =0 =0 then m l = 0 hence L = 0 and L Z=0 For l =1 =1 then m l = -1,0 or+1 hence L=
and LZ = - , 0, or +
For l =2 =2 then m l = -2,-1,0,1 or 2 hence L =
and L Z = -2 ,-
0, 0, + or +2
For l =2 =2 graphically we can represent it like this:
There are a couple of important commutator relations to remember with angular momentum.
This means that we can know both the total angular momentum and one component of the system. But what about knowing two components of angular momentum...
This shows that we cannot know more than one component of angular momentum at the same time as the commutator is non-zero. This is why we can only know one component and it is by convention that we choose the Z – component.
Ladder operators
We can simultaneously know the eigenvalue of
and
. This is because they commute. But we can raise or lower
by using two special operators, the creation operator (to raise):
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Quantum Mechanics and its Applications
And the annihilation operator (to lower):
These raise or lower the eigenvalue of
hence gives us the next or previous value of the
component of angular momentum. They create or destroy quanta of
.
Matrix representation From equation (35) the eigenvalue equation, we can represent it as a matrix equation.
Where the first matrix is the Hermitian operator (Hermitian matrix), the second and last equation is the eigenvector and q is the eigenvalue (basically a scalar) Simply put:
The is also a technique called taking the Hermitian of a matrix which ca n be achieved by first:
Take the transpose of the matrix in question
Then take the complex conjugate of it
The final thing about matrices to mention is a group of matrices called the Pauli spin matrices. These are particularly useful when it comes the angular momentum.
These have corresponding operators called
and
these take the form as follows:
These commute much like the angular momentum operators operators we’ve met before:
Finally there are useful matrix multiplications to remember (often been an exam question!)
For (65) and (67) then other pairs follows the same suite like the angular momentum operators.
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The eigenvalues of ALL 3 spin matrices are +1 and -1 and the corresponding eigenvectors are as follows:
Zeeman Effect
The Zeeman Effect is the observation of the splitting of spectral lines when there is a presence of a magnetic field. Orbiting electrons have an associated magnetic moment which interacts with an applied field which leads to the modifications in energy of the levels seen. It lifts the degeneracy of the ml levels. This is easily described with some diagrams...
There are two important selection rules which govern the allowed transitions seen.
Change in orbital quantum number ( l ) is ±1
Change in magnetic quantum number (m l ) is 0 or ±1
These rules come to light due to the fact that angular momentum must be conserved and photons can only carry angular momentum in quanta of
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Spin Orbit Coupling/ Anomalous Zeeman Effect This was first observed in the Stern-Gelach experiment. This was a search for the quantisation of m l .
Silver atoms Heat source
Inhomogeneous magnetic field
The magnetic field splits the spin states up which are
(furnace)
detected on the screen
It was predicted that there would be an odd number of beams as: Ml = -l......0......+l hence hence always an odd number of states But in fact an even number was observed, with equal and o pposite deflection. This implies that there is spatial quantisation but with ½ integer quantum numbers i.e. ± ½. Since silver atoms have full shells except 1 electron in an S- orbital, i.e. l = 0 because n=1 then this cannot be due to the m l numbers because there is no orbital angular momentum. It was in fact due to intrinsic angular momentum otherwise known as spin! (Interestingly enough, it is not physical spin, like the earth or a football spinning on an axis, this is because it would break the speed of light which is naughty). The electron spin interacts with orbital angular momentum in all shells except for the S shell where there is not orbital angular momentum (remember the quantum numbers). They effectively cause an intrinsic magnetic field which lifts the degeneracy of the m l numbers, this is known as spin orbit coupling.
Proofs Here is a shed load of proofs and derivations which I made during my exam revision you should find them useful and I found that many came up in exam papers.
Proof that eigenvalues are orthogonal We know from equation (39) that:
And by definition we know that:
For two measurements of some operator we have:
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Quantum Mechanics and its Applications
Ok so the trick here is to:
Take the complex conjugate of all of 1. And then multiply by Ψ2
Multiply 2. By Ψ1 which leads us to the following...
*
Remember we can write 1. As (because of the definition stated earlier)
Now we integrate each equation 1 and 2 over all space to obtain:
Now notice, from the first equation, if we call f=
and g=
we fine that the two left hand sides are
equal! And so the two right hand sides must also be equal! So we are left with...
And
The second equation we can rearrange quite simply as we are integrating o ver the same limits and have a common factor of
:
*
Note! I have used E 1 and not E1 this is because we know that the eigenvalues of Hermitian operators
are...yes, real numbers hence taking the complex conjugate of a real number doesn’t change it. We can conclude from this that if:
It is these two last l ast statements which show that the eigenvalues are orthogonal.
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Quantum Mechanics and its Applications
Proof that 2 observables have common eigenfunctions and they commute These two bits of proof go together rather nicely. If we have two observables (variables) then for a set of wave functions Ψn we have:
and
But with a bit of a leap of faith we proceed as follows:
Then be careful to make clear that we are distinguishing that the following has a scalar:
So we have produced another eigenvalue equation:
Only differs from
by some constant so we arrive at a similar equation as to the top
equation, which have a common set of eigen functions,
.
Now to show that for such a variable, they must commute...we re-write the wave function combination of states (like I said before this is a useful first to many proofs).
as a
Now start with a commutation:
This isn’t too hard once you remember to operate like normal to the right of the operator, and replace it with the eigenvalue corresponding c orresponding to the operator, and remember we can take the a n out in brackets as it is just a constant and is not being operated on.
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Quantum Mechanics and its Applications
Proof that [Lx, Ly] = i Lz This proof isn’t too hard to remember, I find it easier if you start from first principles and working it out rather than purely remembering steps, this is how I do it: We note that:
We calculate angular momentum by the cross product of position and momentum; from this we can calculate the resulting components from a simple 3 by 3 matrix like this:
Now you may notice that each component is the angular momentum in each direction, i.e. L x , Ly and Lz. Now let’s do the commutation:
I know its a bit daunting now, but now you have to multiply it all out to yield:
Now this is important, the 4 terms which have been highlighted contain non-commuting components and so cancel with each other. So now our commutation has been simplified to:
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Quantum Mechanics and its Applications
Is an operator Hermitian? This is a way of determining or showing if an operator is Hermitian. It all starts from one of the conditions of being Hermitian, equation (39):
Take the operator you’re using so let ’s say it’s the angular momentum operator given by:
Keep f and g as they are, just some wave functions, and replace the operator operator.
with our new
We start simplifying the right hand side first as follows:
Now carry out the integral using the integration by parts method:
The first part of the solution is equal to zero so we are left only with the last integral part. Hence:
But following out first equation this must be equal to the left hand side so that so we can simplify the above as so:
Notice that I have put it all inside the integral to make the angular momentum operator appear more obvious, so now we can take out the rest and replace with just our operator symbol
:
This, except for me switching the left and right around, is identical to the top fi rst equation
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Quantum Mechanics and its Applications
Expectation values Here is how to show that the expectation value is given by:
Proceed as followed...
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Quantum Mechanics and its Applications
Density of states survival guide! I found much information on the internet about this as I didn’t follow this in lectures well but if you can remember a few bits from below it is MUCH easier. Remember this is part of the application side of things so it would be easier to read that then come back to it . The only thing to keep in mind is what dimension we are working in, 1, 2 or 3 dimensions.
The volume of our “box” which can contain our states is given by:
The volume of our Fermi surface given in K-space is calculated by:
Still working in K-space, our density of states in K-space is given by:
To convert this into terms of energy, g(E)dE we have to do 2 things: We know that K is...
Now take our calculated g(K)dK can sub the above into K and dE and we obtain results for the density of states as follows:
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Quantum Mechanics and its Applications
Applications of Quantum Mechanics Mechanics Introduction So this is the application side of the module, firstly however I will go over a little more theory which is required.
Pauli Exclusion Principle and fermions “No two identical fermions can occupy the same quantum state”. This the main point of the principle, this is why only two electrons can occupy each state but have different spins, and from our quantum numbers this means that no two fermions can have the same 4 quantum numbers. Fermions are particles which have half integer spin, these include electrons, positions and protons. A nice way of working out if something is a f ermions or a boson is to count up its components, if there are a odd number of components then it is a fermions, if it is even, it is a boson. For example, a proton contains 2 up quarks and a down quarks = 3 components therefore is a fermion whereas a oxygen atom has 8 protons, 8 neutrons and 8 electrons = 8(3 quarks) + 8(3 quarks) + 8 electrons = 56 = boson.
Metals and Free Fermions Molecules form due to this simple fact. Atoms want to be stable, and they want to have lower energy therefore more stable. The can achieve this by sharing electrons (bonding) which lowers their kinetic energy as the area a electron can be is increased and so an electron will have a larger wavelength:
Hence, if wavelength increases, the kinetic energy decreases, more stable. This is an example ex ample of covalent bonding. Ionic bonding happens with two ions with opposing charges, most typical example is Sodium +
-
Chloride, (or any salt really) which has two ions Na and Cl these are electro-statically attracted together. Metallic bonding happens with metal ions like copper which together are held by the delocalised of electrons from their outermost shells, which are free to move by confined to the metal as a whole (due to the potential energy of the metals and electron).
Fermi Wave Vector The wave vector of the highest occupied state is called the Fermi wave vector. So for example a system with 10 electrons in, the electron which is in the highest state will correspond to the Fermi wave vector of that system. Similarly the energy of the highest occupied state is called the Fermi energy.
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Quantum Mechanics and its Applications
Density of states The number density of a system for example the electrons in a bar of metal will be calculated as followed:
The number density is equal to the total number of electrons dived by the length of metal. We know for such a system the Fermi wave number K F if given by;
Well the total number of electrons in the system must be equal to:
The multiple of 2 comes about from spin up and spin down states so we have 2 lots. The 2 lots of n F is because we have to count both the + wave vectors and then – wave vectors. And the addition of +1 is due to the ground state electrons (remember we only add +1 because we have taken then factor of 2 out). When 2 distinct states (like two states with different quantum numbers) have the same energy, they are said to be degenerate. For example, an S orbital can have two electrons electrons in but with the same energy, this is two-fold degeneracy. In a very large (classical) system we would expect a continuum of energies because, quite simply our technology cannot measure the quantisation. We would write the difference dif ference between one energy, E and then next as E + dE, then the system has g(E)dE states, where g(E)dE is called the density of states. In general for arbitrary E:
From the above we can compute quantities of interest: Total N is given by:
Total energy, E is given by:
Average energy can now be computed: Page 23 of 28
Quantum Mechanics and its Applications
Conserved densities
From basic conservation of energy the flow of matter through a surface, S must be equal to the
negative rate of change of what’s inside (volume, V) hence:
j is current density and p is matter density. density.
It follows that we can show that in 3-D the current density is given by:
Magnetic Susceptibility
In a magnetic field we observe that system shifts energy because of the spin up and spin down electrons. The amount it shifts is given by:
Where B is the strength of the magnetic field and μ is the Bohr magnetron. This leads to a difference of energy between the electrons hence:
Finally yielding two equations:
The Nucleus and Binding Energy
Protons and neutrons are bounded together to form a nucleus. There are some important observations with this: Nucleon density is almost constant across all nuclei:
However the energy which holds the nucleons together, the Binding Energy is hard to quantify, there is a semi-empirical formula for binding energy, B:
This looks quite horrible but i will explain the justification for each part. Page 24 of 28
Quantum Mechanics and its Applications
The The
part is the volume term. The nucleus is attracted by strong force to their neighbours part is the surface term. Nuclei near the surface have fewer neighbours and so we need to
correct for this. The
part is the Coulomb force term. This term compensates for the fact that protons repel
each other. The
part is the Asymmetry term. This is i s from treating the protons and neutrons as free
Fermi gases. The
part is the paining term. Observed that having a odd number of either electrons
or neutrons cost an energy which is proportional to A-
4/3
.
All this together gives us our binding energy curve.
Insulators and Semiconductors If the Fermi energy of a system lies between a gap in the density of states there are:
No low lying excitations
Cannot conduct electricity
No contribution to heat capacity
Transparent (photon wavelength invisible to our visible visible light spectrum)
In general, anything with a band gap of approximately > 2eV is an insulator.
The diagram on the left shows the Fermi energy is between occupied and unoccupied states whilst the diagram on the right shows how an occupied state electron could be excited into a n unoccupied state by a photon, it also shows that the photon must have the correct energy to overcome the band gap for this excitation to happen. This gap comes from the energy gained by the delocalisation of the electrons in ionic bonding. In covalent solids, the gap comes from the scattering off the lattice of positive ions which create periodic potential wells.
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As for semiconductors (C,Si,Ge)...
Core electrons are tightly bounded to the nucleus, unaffected by the rest of the crystal
Valence electrons are loosely bounded and ar e free to move
You can see the conduction band at the top and the valence band at the bottom, notice the Fermi energy is in the middle of the band gap
We can dope these semiconductors to change their ability to conduct. There are two types of atoms which we use to dope semiconductors with. These are group 3 elements and group 5 elements. Group 3 elements are called acceptors, ac ceptors, these take electrons away from the semiconductors and cause holes, which acts as positrons (anti electrons), this causes an energy band just above the valence band. We call these semiconductors, P – type. Group 5 elements are call donors, these give electrons to the semiconductor and cause an energy band which is just below the conduction band. We call these semiconductors, N – type.
Standard Model We can only use quantum mechanics to help understand the particles in the standard model, but not to describe it... We know the operators for energy and momentum is:
If we use these in place of the usual description of energy then we find:
And then:
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Klein Gordon Equation We first quantify free particles, so like above we use the operators in place of the normal energy equation:
Which then becomes using (87) again:
I introduce a concept called scaled variables, we take the positions, x, y and z, along with time, t and rest mass, m 0 and scale them to obtain, x’,y’ and z’ with t’ and m0’. These all have new units of powers of energy, see below:
Using these and by dropping the prim es ‘, we arrive at the Klein – Gordon Equation:
Few points to make:
Correct description for spin less bosons
Schrödinger tried this equation first but rejected it as it was “wrong” for the energies of the hydrogen atom.
Dirac equation Dirac suggested looking for a first order equation to describe free electrons. Note that we set
.
If we now quantise it:
Where α is a vector and β is a constant.
Space and time appear as derivatives. It should be possible to make this theory consistent with special relativity.
Hope to explain the origin of spin, if so, Ψ will have to have components spin up and spin down.
Still must be consistent with the Bohr correspondence principle:
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Quantum Mechanics and its Applications
So what exactly are α and β?
They have the following form, where they are made up of the Pauli spin matrices we discussed earlier.
But it doesn’t only have to be the
it can be any of the 3.
The negative eigenvalues from these predict the existence o f antimatter. Dirac then went on to help invent Quantum Field Theory which subsequently became the standard model.
The nuclear force The final bit to cover is a little l ittle theory on the nuclear force.
With the standard model, all the forces are the result of particle exchange
Electromagnetic force is mediated by photons
Weak nuclear force is mediated by the W and Z bosons
Strong nuclear force is mediated by gluons
±
0
So the last thing to mention is that there is a measure of the coupling between charge and electromagnetic field. We come to this from considering the potential from the coulomb interaction:
α is
dimensionless, and if we use another scaled variable, in this case scaled charge:
α
is the measure of the coupling, we can calculate it like this:
So there you have it, this is pretty much it, shame to say that this guide doesn’t cover everything but should provide you with a good starting point or a nice aid , remember to learn the proofs! Good luck.
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