Long explanation about your questions
Nima said:
Hello,
I have suffered from the same problems that you describe. I think that quantum physicists and theoretical physicists in general often do not (or even cannot) make a distinction between Mathematics and Physics (… sometimes this is difficult). I believe that this is one of the reasons why people have such great difficulties to conceptually understand the subject.
So, I will try to give answers in a way that I would have hoped for.
I will start out very profound and hope you will not be offended if I say certain things that are very basic and that you know already. It will hopefully provide a framework where the complicated things will make easier sense to you.
1.) Wave Function
In 1924, Louis de Broglie suggested that all particles can be considered waves.
He gave a derivation in his PhD dissertation where the major formula was
lambda=h/p
where lambda is the wavelength of the particle, h is Planck´s constant and p is the momentum of the particle. The momentum in turn is p=m*v where m is the mass of the particle and v is the speed of the particle. The formula lambda=h/p
is correct for slow particles. For fast ones you must use the 'relativistic formula' since according to special relativity, the faster a body moves, the more mass it gains. I will not go into that here, but see my answer to 3) below.
As you know, when you use mathematics you think in "functions". In high school you learn 'y is a function of x'. For these functions, you can take the derivative dy/dx as the first derivative and d^2y/dx^2 as the second and so on.
In Physics, the most important derivatives are the first and the 2nd derivative and in particular the 2nd derivative. You may ask, why is that? The reason is related to Newton's law "Force = mass * acceleration". Typically you have a given force in physics and you want to describe the movement of bodies in this force field.
A consequence of this "I have a force and now I try to predict the movement of bodies in its influence" is that you end up with 2nd order differential equations. One such equation is the classical wave equation which describes the movement of a wave on a string
delta^2/delta x^2 [y] - 1/v^2* delta^2/delta t^2 [t] = 0
x is the x coordinate, t is the time, v is a constant and gives the tension of the string and the mass per unit length and
y is the wave function!
A general solution of this wave equation is
y=f(x-v*t)+g(x+v*t)
where y is the wave function.
Here, I believe is an example why physics (and particularly modern physics) is so difficult to understand. All of a sudden, we use a mathematical term rather than a physical term.
The wave equation is a second order, partial, differential equation and the solutions to such equations are functions, rather than variables.
Erwin Schrödinger tried to find a model of the behavior of the electron in the hydrogen nucleus and because he knew that electrons behave like waves he tried to mathematically model the hydrogen atom as a wave system with a wave equation.
His wave equation was
i*hbar*delta(psi)/delta(t)=-hbar^2/2*m*delta^2(psi)/delta^(x)^2+V*(psi)
Now, if you find a psi that is a solution to this equation then you have a wave function for that equation. The Wave Function Psi is a Solution to this Equation, in the same way as the y above is a solution for the classical wave equation.
The term function comes from the mathematics for second order differential equation and the term wave comes from the physics that the electron has wave character.
2.) Intrinsic Angular momentum
= spin. This is a physical term. The intrinsic (non-orbital momentum) of a particle is called spin. Think of the Earth going around the Sun. The momentum around the Sun is orbital momentum. However, the Earth also turns around its own axis, ie. the Earth spins and has a spin.
The electron rotates (this is not really true, but fine enough as a model) around the atom and therefore creates orbital momentum. However, it also has an intrinsic spin, ie. rotates about itself (again, this is probably not really true but as a model, it is fine).
Spin was discovered in the Stern-Gerlach experiment. Particles are grouped into two types relating to spin. The value of the spin is either half or integer multiples of hbar, ie. 1/2 hbar, 3/2 hbar or 0 hbar (no spin), 1 hbar, 2 hbar. The half multiples are called Fermions, the integer multiples are called Bosons.
3.) Relativistic
='Physics slang'. Results of Einstein's Special Relativity show that mass and energy increase, length contracts and time slows down the faster you move (up to the speed of light). When physicists say relativistic then they usually mean that relativistic effects must be taken into account in a calculation. Most things (but not all) that move less than 10% of the speed of light are not considered as relativistic. The most important mathematical term is the factor k=square_root_of(1-v^2/c^2) where v is the speed of the quantity in question and c is the speed of light. If you use v=0.1 and c=1 (ie. v=10% of the speed of light) then the k term becomes k=square_root_of(1-0.1^2/1^2)=0.995 of the non-relativistic value, so there is almost no difference.
For making quantities smaller, multiply by the k term, for making them larger divide by this term. For example, a 1m rod that moves with 3/5 the speed of light is
1m* square_root_of(1-v^2/c^2) = 0.8m long
4.) The Coupling Constant
Mathematical concept. Describes a number (=constant) that gives the force between two particles where these two particles interact via one of the fundamental four forces (=electromagnetic, strong force, weak force, gravity).
Most often, however, Coupling constant refers to the electromagnetic force between two point charges such as, for example, two electrons.
The quantum mechanical expression of the potential energy between two point charges is
V=((1/(4*pi*epsilon0*hbar*c))*e*e)/r
epsilon0…permittivity of free space, a constant
e…..electron charge 1 = 1.602x10^-19 C
e….electron charge 2 = 1.602x10^-19 C
r…..distance between the charges
hbar… Planck´s constant divided by 2*pi
c….. speed of light
Except for the distance r, every term in the expression is a constant and if you put the numbers in then you get approximately 1/137 = (1/(4*pi*epsilon0*hbar*c))*e*e. This is called the coupling constant of the electromagnetic force. It is also called fine structure constant (same thing).
5.) Spin is parallel/anti-parallel to momentum
Physical and mathematical: parallel means that physically the Spin and angular momentum are aligned in the same direction and mathematically that you add them, just like two vectors. anti-parallel means that physically the Spin and angular momentum are opposite in their direction and mathematically that they must be subtracted, ie. the difference of the vectors taken.
6.) Charge Multiplets
Mathematical concept that describes consequences of isospin in Nuclear Physics. This is fairly advanced. To really understand it, you first must understand eigenfunctions and their relationship to quantum mechanics and spin operators. This is one of the most difficult things in quantum mechanics and takes some time.
I will briefly try to explain. In 1932, Heisenberg (yes, the famous one) suggested that certain groups of particles such as the nucleons, ie. the proton and the neutron, could be considered as the charge state of the same particle. As you probably know, the proton has a positive charge, whereas the neutron is, well, neutral.
Heisenberg suggested that both particles have an isospin of T=1/2 but different 'projections' T3: T3=+1/2 for the proton and T3=-1/2 for the neutron. This is where it becomes difficult. These projections are eigenvalues of an operator, the isospin operator, which acts in the same way as the spin operator.
Formally, the isospin is treated as quantum mechanical angular momentum and calculated with in the same way. Isospin can be used as a quantum number to label isobars (=nuclei with the same atomic number A), where to each nuclear state an isospin T with third component T=1/2(Z-N) is added. Z is the number of protons and N the number of neutrons.
Finally, how is all this related to Charge Multiplets? I show an example.
I use the following notation: CL [37 17] means the nuclei CL (= Chlorine) with mass number A=37 and proton number Z=17. It has 37-17=20 neutrons and hence N= 20.
The following nuclei CL [37 17], A [37 18], K [37 19] and Ca [37 20] have all the same atomic number 37 but different number of protons and neutrons.
Using the formula T=1/2(Z-N), their respective isospins are T3=-3/2, T3=-1/2, T3=+1/2, T3=+3/2.
They form what is called a Charge Multiplet. They have the same mass number but different charges which is expressed as the isospin.
7.) Supermultiplets
Mathematical concept, similar to 6, but related to characteristics of quarks.
8.) Isospin
See 6 above.
10.) What is the Pauli exclusion principle
This is a mathematical concept with physical effects.
It says that no two electrons in the same atom can be in the exactly same state, ie. can have the same quantum numbers n,l,m,s. If n,l,m are the same then their spins must be different. The principle has also a wider application in nuclei where it is partly 'responsible' for certain aspects of the shell model of nuclei and the existence of the so-called magic numbers. Since neutrons and protons are Fermions (spin 1/2 particles) they also obey Pauli´s principle.
Pauli derived it from mathematical considerations which are related to the symmetry requirements for the wave functions.
Cheers,
Roberth
PS: Another reason why people find physics difficult is because it is difficult!
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