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...
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...
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 =...
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...
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...
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...
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...
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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...
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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...
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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...
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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...
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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...