# 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...
11. ### Symbolic computation of gradient

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...