Homework Statement
A beam of spin-1/2 particles scatters off of a target consisting of spin-1/2 heavy nuclei. The interaction between the particle and nucleus is given by $$V(\vec{ r})= V_0~\delta (\vec{r})~ \vec{S}_1. \vec{S}_2$$
1) Averaging over initial spin states, find the differential...
Let ##\Lambda## be a Lorentz transformation. The matrix representing the Lorentz transformation is written as ##\Lambda^\mu{}_\nu##, the first index referring to the rows and the second index referring to columns.
The defining relation (necessary and sufficient) for Lorentz transforms is...
Say, we have two Hilbert spaces ##U## and ##V## and their duals ##U^*, V^*##.
Then, we say, ##u\otimes v~ \epsilon~ U\otimes V##, where ##'\otimes'## is defined as the tensor product of the two spaces, ##U\times V \rightarrow U\otimes V##.
In Dirac's Bra-Ket notation, this is written as...
Say, we have two orthonormal basis sets ##\{v_i\}## and ##\{w_i\}## for a vector space A.
Now, the first (old) basis, in terms of the second(new) basis, is given by, say,
$$v_i=\Sigma_jS_{ij}w_j,~~~~\text{for all i.}$$
How do I explicitly (in some basis) write the relation, ##Uv_i=w_i##, for...
I have offers from UMass Amherst Physics and Case Western Astronomy for their respective Phd programs. I'm having a tough time deciding between the two. Leaving aside the differences due to astronomy vs physics, what other factors are important to consider while making a decision?
UMass Physics...
Homework Statement
A system in thermal equilibrium at temperature T consists of a large number of subsystems, each of which can exist only in two states of energy and , where . In the expressions that follow, k is the Boltzmann constant.
For a system at temperature T, the average number of...
A system is in contact with a reservoir at a specific temperature. The macrostate of the system is specified by the triple (N,V,T) viz., particle number, volume and temperature.
The canonical ensemble can be used to analyze the situation. In the canonical ensemble, the system can exchange...
Say, I have two spin-1/2 particles in the states characterized by ##(n=2, l=1, m_l=1, m_s=1/2)##and##(n=2, l=1, m_l=1, m_s=-1/2)##. Now, using something like the jj coupling scheme, I first find out the (orbital+spin)angular momentum for the individual particles:
(i) ##| 11\rangle...
All textbooks and material that i've read on the topic state that the deuteron being a weakly bound system, has no excited state. They also go on to state that the deuteron exists as a mixture of ##^3 S_1## and ##^3D_1## states.
So, are these states degenerate in energy? That is, are both of...
The state, ##| S\rangle##, say, of a system is represented as a vector in a Hilbert space.
##\psi (x, t)## is the representation of the state vector in the position eigenbasis; ##\psi (p, t)## in the momentum eigenbasis et cetera. That is, ##\psi (x, t) = \langle x|S\rangle##; ##\psi (p, t) =...
Homework Statement
A parallel beam of light of wavelength ##\lambda## is incident normally on a thin polymer film with air on both sides. If the film has a refractive index ##n>1##, then, for what value of the thickness, can second order bright fringes be observed in reflection?
Homework...
Suppose a change of basis from basis ##B## to basis ##C## is represented by the matrix ##S##.
That is, ##S## is the transformation matrix from ##B## to ##C##.
Now if ##t## is a given linear transformation, ##t:~V\rightarrow V##, with eigenvectors ##\epsilon_i##, say, and ##T## is the...
Suppose we have a charge, ##q## and a magnetic dipole moment, ##\vec m##. They don't move, nothing changes with time, in short, a static situation.
Now, we have at least some regions of space where both the electric field, ##\vec E## and the magnetic field, ##\vec B## are non-zero. That means...
The common explanation for the electric field inside a conductor being zero goes something like this:
Suppose a perfect conductor is placed in an electric field, the external field causes the free charges to redistribute in such a way, that the resulting internal field exactly cancels off the...
What does this statement mean: "Apparent magnitude of a star, X, is m." ?
m_2 - m_1 = -2.5 log (B_2/B_1)
Apparent magnitudes are defined relatively, right? We can talk about differences in apparent magnitudes. If i know the ratio of the brightnesses, i can find out the difference in...
How do i derive the Dirac equation from L_{dirac} = \overline{ψ}_α [i(γ^μ)_{αβ} - m]ψ_β ?. I can get it for the \overline{ψ} , but i'm having trouble deriving it for ψ .
Homework Statement
Show that the tensor
θ_{ik} = g_{ik} - U_{i}U_{k}
projects any vector, V^{k}, into a 3-surface orthogonal to the unit time-like
vector U_{i} (By a projection, the vector θ_{ik}V_{k}, is implied).
Homework Equations
The Attempt at a Solution
The projection should...
I am looking for pedagogical guides to fitting rotation curves with MOdified Newtonian Gravity. I want to study how to fit the luminosity data as well as the kinematic data. I have studied some published papers on MOND fits(Sanders, Mcgaugh, blok etc). But, they are not sufficiently elaborate...
Let \normalsize S[y] = \int ^{a}_{b} f[y, \dot{y}, x] dx be the functional i want to minimize. Why does \normalsize f (inside the integral) take this specific form?
Would i not be able to minimize the integral, \normalsize S , if f had any other form instead of f = f[x, y, \dot{y}] ?
From what I've understood,
1) the metric is a bilinear form on a space
2) the metric tensor is basically the same thing
Is this correct?
If so, how is the metric related to/different from the distance function in that space?
Some other sources state that the metric is defined as the...
Homework Statement
A canonical transformation is made from (p,q) to (P,Q) through a generating function F=a*cot(Q), where 'a' is a constant. Express p,q in terms of P,Q.
Homework Equations
The Attempt at a Solution
A generating function is supposed to be a bridge between (p,q) and...