Search results

1. Concise and complete textbook on non-relativistic QM

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...
2. Higher mathematics learning techniques

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...
3. Grad-school chances looking slim: how can I improve them?

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...
4. Changing the order of integration: surefire method?

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)}...
5. Computing luminosity from surface brightness

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 =...
6. Projecting an abstract state onto position/momentum/energy spaces

Homework Statement Consider the quantum harmonic oscillator in the state | \psi (t) \rangle = \frac{1}{\sqrt{14}}\left( 3 | 0 \rangle \exp{\left( -\frac{1}{2}i \omega t\right)} + 2 | 1 \rangle \exp{\left( -\frac{3}{2}i \omega t\right)} + | 5 \rangle \exp{\left( -\frac{11}{2}i \omega t\right)}...
7. Computing Potential of a Spherical Shell w/o using Newton's theorems

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...
8. Expectation value of the square of the observable

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}...
9. Expectation value of an operator (not its corresponding observable value)

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...
10. Galaxy rotation curve: Applicability of formula

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...
12. Forming an orthogonal matrix whose 1st column is a given unit vector

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...
13. Cause of Plastic Container Deformation in Microwave Oven

Greetings, Earlier this afternoon I managed to ruin a perfectly good plastic container in a microwave oven by unwittingly deforming it. I curious as to why it happened. All I did was warm it (and the chilli inside it) for about a minute with the lid firmly secure. I've been told this was a...
14. Algorithms for quantifying intersections of subspaces

Greetings, I'd like to know how one goes about finding a basis for the intersection of two subspaces V and W of a given vector space U. I am aware of the identity V \cap W = (V^{\per} \cup W^{\per})^{\per} (essentially the orthogonal space of the union of orthogonal spaces of V and W), but this...
15. Strange resistor voltage curve in an RC

Hello, My data gravely perplexes me. I've set up a simple series RC circuit with an AC emf of 14V @ 20Hz. Resistor and capacitor values are 3.9E4 ohms and 4.7E-5 farads respectively. Ideally, the voltage across the resistor should be a simple sine curve, since this is basically an RLC series...
16. Transformer efficiency at low (and high) frequencies

Hello, I've been thinking about why (elementary) transformer efficiency drops drastically at very low frequencies. I know hysteresis effects play a major role in reducing efficiency at high frequencies, but why low? I realise that as we reduce the frequency of the emf, we're making the...
17. Yahtzee Trouble

Hello, I'm confused as to what is the probability of rolling 4 of a kind in one roll, in Yahtzee. I'm thinking its simply C(6,2) * C(2,1), which gives a probability of \frac{30}{6^5} = \frac{5}{1296} . However, most of the sources I have seen regarding this outcome give a probability of...