Campuses:
Return to Q&A main page: Q_A
Q&A for the previous lecture: Q_A_1125
Q&A for the next lecture: Q_A_1202
If you want to see lecture notes, click lec_notes
Main class wiki page: home
To obtain a total antisymmetric wave-function for a fermion, we have to have the spin or spatial wave function be antisymmetric when taking the product. So my question is, what do we do when we have a coupled spin or spatial wave function? How do we distinguish what is symmetrizing and antisymmetrizing parts in this system?
what is the difference between the singlet and the triplet states in terms of a physical picture? especially what is the physical difference between the up-down state of the triplet and the singlet state?
I'm also confused by this.
A spin triplet is a set of three quantum states of a system, each with total spin S=1 as defined. Similarly a singlet state is a quantum state of a system in which we are given one state with a total spin S=0 as defined similarly in the text. The physical picture you ask for are the configurations that make up a total spin of 1 and 0, which you can find by operating with S^2.
Now you probably saw what the configurations were that made up these triplet and singlet states. This is the actual physical picture of what is happening. Now I think I know why you are confused. What does it mean to have 3 states with the same spin? Well it just means we have some probability of having a certain configuration, which has total spin 1. With our singlet state we only have one choice that makes the total spin 0, which we call the singlet state.
The physical difference between the up and down state is just the spin it has, up being 1/2 and down being -1/2. Maybe I didn't answer your question though and help you find what you are looking for. I feel that this is the toughest topic in this book, so I wouldn't be surprised if I didn't quench your thirst.
Can we perhaps get another count of Pre-lecture Q/A points?
Perhaps…
On page 244, Griffiths mentions that he multiplies the degeneracy dk by 2 because of the spin. When we are talking about photons, I usually hear that this factor of 2 comes from the polarization of light, and the fact that it photons can only travel transversely. What is the story? Did we first think it was 2 when we working semi-classical(i.e. before modern QM) and then discovered spin, and accounted this factor of 2 because of spin in modern QM?
The blackbody spectrum was discovered by Max Planck around 1900, I think it was either exactly 1900 or in 1901. Einstein's quantization of light came out in 1905, so it had to be discovered before light was considered a quantum particle/photon. Now there is a very lengthy debate about if light is a particle or wave going back to Newton and Hyugens, but it was mostly resolved before 1905 and considered a wave. I believe some of the early experiments that led people to think of light as a wave instead of a particle could actually show that light is polarized, which is much easier to explain from the wave theory than the particle theory. You can even show polarization of light with a clever trick and two expensive enough pairs of sunglasses. So I would say we knew it was polarized pre-quantum physics and that Planck most likely used a factor of 2 in 1900/1901. When the Stern-Gerlach experiment led to the foundations of spin it must have been a relief when it worked out that the spin of a photon could lead to two polarizations.
how did griffiths get the 1/5 and 3/5 values for the coefficients in the last equation of page 187??? i see it in the graph on page 188 but how would one calculate these values if the graph was not available??
It seems like you use 4.185 to get the general states with some undetermined coefficient in front, and Griffiths just lists all of the coefficients for a bunch of different cases in the table. Intuitively I would have to say that you need find the exact numbers (fractions in terms of hbar) by solving some eigenvalue/eigenvector problem. It seems as if the upper left most part of the table corresponds to the singlet/triplet values on page 185 and 186; for any given state you would need to run though the raising and lower operators and figure out what the eigenvalues are for such a state. But I would like to see the coefficients be found for any state and then the values in the tables reproduced; maybe Yuichi can do it in class?
On pg. 185, the singlet equation (eq. 4.178) looks right, but can somebody explain how it was derived?
He finds a state that is orthogonal to all the states in the triplet equations. Luckily, the singlet shown satisfies this condition, since the inner product with each guy in the triplet state would result in 0. Since this is not in the triplet group and there is one missing state in this whole system, this most mean that this is our singlet state, so we find it by taking some general spin vector and taking the dot product and solving the equation that makes our inner product. Doing this, we find equation 4.178.
Can someone explain where the reduced mass concept comes from? I remember using it in classical and astro, but I don't remember why it is useful or where and how it was derived.
Read this: http://en.wikipedia.org/wiki/Reduced_mass.
It takes into account the nucleus, when looking at the atom, and taking into account the nucleus allows you to see the fine-structure splitting, which is discussed in chapter 6. They simply replace m with mu, and you get a split in the regular energy levels you see.
It depends which particle you treated as stationary. Take the hydrogen atom as an example, for simplicity we treated the nucleus as stationary only the electron is rotating about the nucleus, which is not necessarily true. remember the reduced mass equation is <math>m_r=\frac{m_e*m_p}{m_e+m_p}</math>, mass of electron is negligible compare to proton, thus normally the reduced mass of electron is just m_e, but if we treated the electron stationary and nucleus evolving about electron, things would be different.
Can you explain the fundamental structure and the relation among those components such as mesons, quarks, bosons, and fermions? I just want to understand how they look like and how they connect each other.
I have always found the particle classification as a little confusing, but here is what I know. All particles with half spin (1/2, 3/2, …) are fermions and particle with integer spin (0, 1, …) are bosons. Important examples of fermions are electrons and protons; bosons are things like the photon or He4. The distinction is important because each class has a vastly different statistical distribution; fermions are confined to two particles per each energy state but bosons can have all of the particles in one energy state.
Quarks make up hadrons; hadrons are protons and neutrons. Within the classification of quarks, you can have mesons or baryons. Mesons are a quark anti-quark pairs and baryons are three stable quarks. I'm sure there are very physically different theories based on mesons or baryons, but I am not familiar with them; mostly everything I've heard has been about color, which is similar to charge except instead of having a positive and negative charge you can have 3 different colors.
For a more non-mathematical description of subatomic particles I find wikipedia to be quite straightforward. But the particle classification system as it is now depends on why you're looking at them - are you concerned with energy distribution (fermions, bosons) or are you concerned with constituent parts of particles (ie. hadrons are made up of 3 quarks)? It's not like animal classification where one animal belongs to a single species; every particle is either a boson or fermion, so you can have a hadron that is a fermion or a collection of hadrons that make a boson. As physics typically goes, it is somewhat confusing until you can get an intuitive grasp for the odd way physicists do things.
Dearest Wiki Readers,
A simple, fundamental, question I have is this. How does the full time dependent wavefunction of a particle (Ψ) relate to the spin function (χ)? Is the spin function a sort of separated variable function of the wavefunction?
-Forever yours,
Esquire
I believe so…I think it's basically a linear combination of the individual states of <math>\chi</math> with an <math>e^{iEt/\hbar}</math>, where the E corresponds to the energy of the particular <math>\chi</math> state.
You can have two electrons in the exact same quantum state, except one has spin up and one has spin down. This suggests that you can take the wavefunction Ψ as being “seperated” from χ I suppose, but there are some constraints; only two states are available for fermions at any given Ψ. If you wish to put them together I believe you can just multiply the two, but as far as we've seen you usually only look at one or the other but rarely ever both because they are “seperated.”
I believe the time dependent wave function will look something like this:
<math>\Psi(x,t) = \psi(x)\chi\phi(t)</math>
I know the spin part of the wave function is sort of an “addition” to the x-dependent part of the wave function, it just wasn't included in Chapter 2 because we weren't considering spin at that point.
I think the time dependent wave function for two or more electrons is a combination of their space eigenfunction and spin eigenfunction. Both space eigenfunction and spin eigenfunction will have a time dependent part and can be one of the symmetric states or antisymmetric states. we just have to make the total wave function be antisymmetric for antisymmetric particles like electron, proton… and symmetric for symmetric particles like photons…
Recently, We are dealing with the spin of the particle. It is strange that the result of our measurement has to turn out to be h/2. It let me think about the Postulate of QM. No matter how the wavefunction of the particle is, the results measurement have always to be the eigenvalues of the operators. Why the only possible values are its eigenvalues? It seems that the universe hates non-eigenvaules so much.
I thought I could answer this, but I cannot. I scoured the book, and everything just comes down to the statements on page 106 that a measurement of an observable must yield one of the eigenvalues of that observable. This statement comes with no explanation, but Griffiths specifically states that this is unlike anything in classical physics. I guess it's just one of those things in QM that you have to accept.
I guess you can say that this is a fundamental postulate for QM. When a particle is trapped in an infinite square well, momentum (as well as the energy) can take only certain values, which is the eigenvalues of the momentum operator for the particle. i.e. <math>p=\pm\frac{n\pi}{L}\hbar</math>. The spin measurements that Hardy is talking about is no different from this case. Or doesn't this analogy make any sense?
How is <math/> p^2=1</math>??? i dont understand what he means by that. and how is its eigenvalues +/-1?
Where do you see this?
Oh sorry, page 205-the exchange operator
Consider how operators can be treated as matrices and multiplied. When you multiply once by P, you interchange the two particles. If you multiply this product by P (obtaining <math/> p^2 f</math>), you interchange them again, which brings you to the same result that multiplying by 1 would have. For the eigenvalues, consider the matrix that transforms a vector (a,b) to (b,a). It is easy to see that this matrix has eigenvalues of +/-1.
There is just one thing that concerns me slightly. When dealing with the sum of the states it says that <math>J_z = L_z + S_z</math> as implied by J = L + S. Does this also mean that <math>m_j = m_l + m_s</math> or do these z components only deal with the total m value?