Thanks, I had never heard of Dirac's book until now. By "complete", I mean self-contained, so the reader does not have to refer to external sources due to the author imprecisely glossing over important topics. Landau, Shankar and Ballentine are complete/comprehensive. Griffiths, Scherrer and...
I've searched high and low for a terse (yet complete) introduction to the foundations of non-relativistic QM. Shankar is unparalleled in terms of completeness, yet it is infuriatingly verbose. Landau's presentation is a bit dated and difficult to follow in many instances. It is also not as...
Math is indeed hard, mathwonk. I suppose one cannot substitute for intrinsic ability.
Thank you for the excellent advice, maze & quasar987. Considering special cases is invaluable for getting the gist of things, but I find it difficult to reconstruct (or even remember) the more general...
What are some methods of training one's mind to absorb and understand rigorous mathematical texts? I have been facing great difficulty as of late in studying fields like abstract algebra, complex analysis and calculus of variations. These are all fields where I am unable to formulate graphical...
Thanks for the responses.
Dr. Courtney: I'm averse to flipping burgers for http://pictures.pichaus.com/95b70d92c70703496ea27d0c4ca3289315f1775f?AWSAccessKeyId=0K4RZZKHSB5N2XYJWF02&Expires=1207860000&Signature=a9%2BcuJEeSMway7k3BOxa5KokPxM%3D" [Broken]. Getting hooked onto that job will...
I'm a 3rd year Physics undergraduate student at the University of Toronto, and my academic career seems to have hit rock bottom. The grades from my past term were absolutely dismal and I have (consequently?) been rejected by various professors for summer work. Since I am effectively unemployed...
Is there a way to transform the limits of integration for a multivariable integral without appealing to geometrical manipulations? For example:
\int_a^b \int_{y_1(x)}^{y_2(x)} \int_{z_1(x,y)}^{z_2(x,y)} f(x,y,z) \; dz \; dy\; dx \rightarrow \int_c^d \int_{y_3(z)}^{y_4(z)}...
Homework Statement
I'm trying to find the central luminosity per square parsec of a galaxy with central surface brightness I(0) = 15 \; mag \; arcsec^{-1}. I need the answer to be in multiples of the solar bolometric luminosity per square parsec.
Homework Equations
m_1 - m_2 =...
Thanks everyone, I finally figured it out. Converting to spherical coordinates and using the law of cosines to represent the norm of the vector difference is needed.
My apologies. I had originally meant a spherical shell, from which the potential of an arbitrary spherically symmetric distribution can be computed. Gauss's theorem is certainly very useful, but the objective here is to compute the potential using the equation for \Phi only, without applying any...
Homework Statement
I'm well aware of how to compute the gravitational [electric] potential \Phi due to a spherical mass [charge] distribution of radius R by using Newton's theorems for spherical shells. However, how does one find an analytic expression for \Phi without invoking these theorems...
That makes sense. I mistakenly thought there is a difference between \langle Q \rangle and \langle \hat{Q} \rangle, but that may have resulted from an abuse of notation.
Thanks a lot, dextercioby.
Well yes, but that's simply the generalized form of the first equation I posted. So what you're saying is that they are the same? \left\langle \frac{\hat{p}}{m} \right\rangle = m^{-1}\int_{-\infty}^{\infty}\Psi^*\left(-i\hbar\frac{\partial}{\partial x}\right)\Psi \; dx?
Homework Statement
I know how to compute the expectation value of an observable. But how does one compute the expectation value of an observable's square?
Homework Equations
\langle Q \rangle = \int_{-\infty}^{\infty} \Psi^* \hat{Q} \Psi \; dx
\langle Q^2 \rangle = \int_{-\infty}^{\infty}...
1. Problem statement
This isn't a homework question itself, but is related to one. More specifically, I'm computing the time-derivative of \langle x \rangle using the correspondence principle. One side simplifies to \left\langle \frac{\hat{p}}{m} \right\rangle, but what is the physical meaning...
Sorry, maybe it does work. I just tried solving for \langle H \rangle in both Maple and Maxima, and the two CASs give different results. Maple gives a single value, but I think Maxima is unable to simplify the final expression to that value.
EDIT: My mistake. Maxima is able to simplify the...
This is very strange. If we run with the assumption that the time-dependence factor can be tacked on so primitively, then the expected value of the Hamiltonian is a function of time.
I see no other way to derive the time-dependent form of \Psi(x,t) from only \Psi(x,0).
In this case, both cosine and sine forms are eigenfunctions of the energy operator. This is because we no longer have the boundary conditions \Psi(0,0) = \Psi(a,0) = 0.
I would think the complete form \Psi(x,t) is easily obtained from the initial condition by simply appending the...
Homework Statement
Derive and plot the rotation curve of a galaxy with logarithmic potential:
\Phi(R, z) = \frac{v_0^2}{2}\ln{(R_c^2 + R^2 + q_{\phi}^{-2} z^2)}
where R_c = 2 kpc, q_{\phi} = const. and v_o = 200 kms^{-1}. Note that v_c is defined for z = 0 only.
Homework Equations...
I'm wondering if there are any convenient symbolic "shortcuts" (i.e. abuse of notation) that enable one to compute the gradient with respect to a certain vector, without decomposing the computation into the vector's individual elements and differentiating with respect to each element. For...
I realise this is due in about ~1/2 hour, but the second boundary condition is given by:
M\frac{d^2\psi_1}{dt^2} = M\frac{d^2\psi_2}{dt^2} = \tau \left( \frac{d\psi_1}{dx} - \frac{d\psi_2}{dx}\right) (0,t)
EDIT: Ignore this post. The result leads nowhere.
According to my calculations, this is true: \frac{d^2\psi_1}{dt^2}(0,t) = \frac{d^2\psi_2}{dt^2}(0,t)
Do you see why?
Hint: Apply Newton's (2nd?) Law to the central mass and find an expression for the net force on M
Homework Statement
Show that if the vector \textbf{v}_1 is a unit vector (presumably in \Re^n) then we can find an orthogonal matrix \textit{A} that has as its first column the vector \textbf{v}_1.
The Attempt at a Solution
This seems to be trivially easy. Suppose we have a basis \beta for...