n only has physical meaning for positive values because if you evaluate the energy eigenfunctions, you find that
E_n = \frac{n^2 \pi^2 \hbar^2}{2ma^2}
Mathematically, at least, it is unnecessary for n to be less than 0. Going by this equation, E0 is by definition the lowest possible energy...
Homework Statement
In a shallow layer of water, the velocity of water in the z direction may be ignored and is therefore (\dot{x},\dot{y}). We can define the Lagrangian coordinates such that the depth of water h is satisfied by the relations
Given that h = \frac{1}{\alpha} and \alpha =...
Ok, but L is a constant and having amended that, it doesn't make the final integral any more correct. The ln form of the last equation is still clearly wrong.
Homework Statement
Find the capacitance of the parallel plate capacitor with 2 dielectrics below. Given that the parallel plate capacitor has area A = WL and the separation between plates is d.
The Attempt at a Solution
My method is to use integration. First solve for two small...
Homework Statement
For my lab work, I have created a theoretical model that goes something like:
T = \sqrt{\frac{ks^2}{x \sin \theta \cos^2\theta}}
where k is a constant, and the variables to be differentiated are x, theta and s. How do I find the error of T? I can find the errors of x and...
Homework Statement
Two masses, A and B, lie on a frictionless table. They are attached to either end of a light rope of length l which passes around a pulley of negligible mass. The pulley is attached to a rope connected to a hanging mass, C.
We are supposed to find the accelerations...
Count Iblis, if I am getting you right, to sum up your method: the Laplacian of f is a scalar. Then, since scalars are invariant under rotation of axes, the Laplacian is invariant under rotation of axes.
Would it be sufficient to leave it as that, or would I need to prove it a bit more rigorously?
Homework Statement
\frac{dy}{dx} = \frac{cosh x cos y + cosh y sin x}{sinh x sin y - sinh y sin x}
I'm really stuck at this one. I don't even know where to start, but I hope that a substitution (ie u = f(x,y)) might be able to put this in a separable form. Any hints please??
Other roads:
1...
Thanks Chip.
Count Iblis, ok, I understand a), but what's the vector F? (Do you mean nabla f?) After all I only have scalar function f. How do I proceed from there? Thanks.
Homework Statement
A scalar function can be represented as a position on the x-y plane, or on the u-v plane, where u and v are axes rotated by θ from the x and y axes.
Prove that the 2-dimensional \nabla^2 operator is invariant under a rotation of axes.
ie,
\frac{\partial^2 f}{\partial...
Homework Statement
The Attempt at a Solution
Ok, first part, no problem, second part (steady state), solved in another thread. Both are pretty tedious, but doable.
However, I am quite stumped by the third part of this problem (V = V_o sin wt) .
The second part of the problem...
Homework Statement
Question and part 1 as above. The second part involves solving this equation where L = 8R^2 C. The system is kept in steady state by maintaining V(t) = -Q/C (constant). V(t) is then set to 0 at t=0.
It also says "Note that V(t)=0 for t>0 and that appropriate initial...
Homework Statement
I don't have the answer for this one, so hopefully someone can help...
(a) By reversing the order of integration, evaluate \int_0^8 \int_{y^{1/3}}^2 \sqrt{x^4 + 1} dx dy
(b) By evaluating an appropriate double integral, find the volume of the wedge lying between the planes...
Homework Statement
(This is a truncated question.)
The electric field of a circular sheet of charge of radius a and surface charge density sigma and distance x away from the centre of the sheet is
E = \frac{\sigma}{2 \epsilon_0} [1 - \frac{x}{\sqrt{x^2 + a^2}}]
Prove that for x > 0...
I've got an older edition (ca 1995!) Nick. It's weird because Griffiths does a few examples (ie, expand a+ a-, etc) with the operator that I stated at the beginning.
I didn't know about the Hermitian part. But this particular question, which I don't think is in newer editions (I checked)...
Homework Statement
This is problem 2.11 from Griffith's QM textbook under the harmonic oscillator section.
Show that the lowering operator cannot generate a state of infinite norm, ie, \int | a_{-} \psi |^2 < \infty
Homework Equations
This isn't so hard, except that I consistently get the...
Compuchip, unfortunately, I have an old edition without the footnote, and a number of problems are different from other editions.
Ok... this wasn't so hard! I substituted the stationary state wavefunction into the formula for <H> as you suggested, and due to the orthonormality the two middle...
Thanks guys... but I'm still stuck. Based on the wave function (they share the same constant A) I gather that I have an equal chance of my measurement having E1 or E2. I'm going to literally guess the answer would be the average of the two, ie, 0.5(E1+E2).
Since expectation value of x...
Homework Statement
This problem comes from David Griffiths' quantum mechanics book which I have been going through on my own.
A particle in the infinite square well has its initial wave function as an even mixture of the first two stationary states
\Psi(x,0) = A(\psi_1 (x) + \psi_2 (x)
I...
Homework Statement
Two planets A and B are at rest with respect to each other and are L apart in this frame. They have synchronised clocks. A spaceship flies at speed v with respect to planet A and synchronises its clock with A-B.
We know that when the spaceship reaches B, B's clock reads...
I run into another problem once I go further along this question.
If I solve the equation given here, I actually get v = 2c/5 rather than v = c/5 quoted in the book.
Homework Statement
This is an example problem although I don't really understand it.
Train A is moving at 4c/5 and train B moves at 3c/5 with respect to observer C.
Let E1 be the event "front of A passing back of B" and E2 be the event "back of A passing the front of B".
An traveller D...
Electromagnetic Fields and Waves by Vladimir Rojansky. I'm not in college yet, and the book goes through the maths quite thoroughly, so it is a wonderful resource for me. I also like the fact that most of the problems are proof-type questions (personal preference). Would appreciate if anybody...
Homework Statement
This is a frictionless system with the wedge on a frictionless horizontal surface. When the system is released, the horizontal wedge (mass M) with diagonal angle theta moves to the left with constant acceleration a. What is it?
I hope I'm right when I say this is...
Homework Statement
A spool of thread comprises a cylinder of radius R1 and is capped with two disks of radius R2, where R2 > R1. Some thread is tightly wound over the cylinder. The whole spool is laid stationary, side down on the ground and the thread is pulled. The spool rolls without...
Sorry Halls, the closed curve is the intersection of the two equations. That curve is closed and the surface it covers is a circle (of radius sqrt(2) I think) then I'm integrating over the plane.
I don't know how to describe it, but I deduced by imagining that the line slices the sphere given...
Homework Statement
Please help me to check whether I did the right working for this problem. Thanks. The numerical answer is correct but I'm not very sure if the working is correct also.
Find \int y dx + z dy + x dz over the closed curve C which is the intersection of the surfaces whose...
Hi Tim, what do you mean dr^2? There shouldn't be a square in the line integral (F.dr) right?
I think I saw where I got it wrong... I did integration by parts for t cos t and got 0 (between 2π and 0)... then I removed the t cos t / t sin t from the expression. But the integration by parts for...
Homework Statement
For the field \bold{F} = (y+z) \bold{i} - (x+z) \bold{j} + (x+y) \bold{k} find the work done in moving a particle around the following closed curve:
from the origin to (0,0,2π) on the curve x=1-cos t, y=sin t, z=t; and back to the origin along the z-axis. The answer is 2π...
Homework Statement
Write an evaluate a triple integral in spherical coordinates for the volume inside the cone z^2 = x^2 + y^2 between the planes z=1 and z=2.
The answer is 7π/3
The Attempt at a Solution
Substitute values to work out the limits. From z^2 = x^2 + y^2, substitute for...
Homework Statement
Here, I have two ways of finding the y-coordinate centroid of a semicircular arc using polar coordinates.
First one is considering a circle of radius a, centred at the origin. What I have done is \int ds = \int_0^{\pi} a d\theta = \pi a and then \int y ds = \int r^2 sin...
swayze, perhaps it'd be useful to think more simply, ie, what is the relationship of the points between two inverse functions? The question only asks about one particular point, so it's pointless to find out what the entire inverse function is. (Assuming that, since the question asks about an...
Homework Statement
This isn't really a problem at all. Have been working on some problems in Mary L Boas' textbook with some difficulty, but I have managed to solve the majority of the questions eventually. However, I've been stumped at how to arrive at the precise answers quoted in the book...
Mark, thanks so much, that worked. You meant that since x^2 is the inverse function, it is exactly symmetrical about the y=x, so (2, root-2) maps to (root-2, 2), right?
But I hope to be able to solve the original problem as-is. Just changing functions unfortunately won't reveal what was wrong...
Homework Statement
Find the arc length of y=\sqrt{x} from x=0 to x=2.
The Attempt at a Solution
I don't know, this is a nastier integral than it looks. From the substitutions,
s = \int_0^2 \sqrt{1 + \frac{1}{4x}} dx. From doing this over and over again I already know the answer will have...
Homework Statement
A thin rod is 10 ft long and has a density which varies uniformly between 4 and 24 lb/ft. Find:
a) the mass
b) the x-coordinate of centroid
c) Moment of inertia about an axis perpendicular to the rod
d) Moment of inertia about an axis perpendicular to the rod passing thru...
Oh... sorry Dick I got it mixed up, was doing a question on determinant of transposes before this.
det A = det (A-inv) = 1 or
det A = det A(-inv) = -1, hence. Thanks Dick...
Madhawk, I'm not much of a mathematician...
Ok, Dick:
If A and A-inverse are integer matrices then det A and det (A-inverse) are integers because of the Laplace expansion. det (I) = 1, so det (A A-inv) =1
det (A-inv) det (A-inv) = 1, and since both are equal, det (A-inv) must be 1 or -1.
Is this the right reasoning? Thanx Dick.
Halls, I'm not sure if I'm familiar with that equation. Do you mean the adjoint matrix of A, ie, A-1 = Adj (A) / det A? Also not very sure about your question, what do you mean by A is +1 or -1?, the determinant?
If det A = 1 or -1, then the adjoint = inverse.
I have thought of something, not...
Homework Statement
A is an invertible integer matrix. Prove that if det A = 1 or det A = -1, then the inverse of A is also an integer matrix.
Also prove the reverse, if A-inverse is an integer matrix then its determinant is 1 or -1.
Homework Equations
I'm not too sure how to start...
Homework Statement
For the twisted curve y^3 + 27 axz - 81a^{2}y = 0, given parametrically by
x=au(3-u^2), y=3au^2, z=au(3+u^2)
show that the following hold
\frac{ds}{du} = 3 \sqrt{2} a(1+u^2), where s is the distance along the curve from the origin
the length of curve from the origin to...
Thanks a lot Halls and gabba, I've got it. The matrix equation you were talking about doesn't display though.
Please look through my work here, hopefully it's right:
\bold{E} + \bold{\dot{r}} \times \bold{B} = (E - \dot{z}B) \bold{i} + \dot{x}B \bold{k}
as discussed above.
a) m\ddot{x} = qE...
I did get at the second equation you mentioned, gabba, but still I don't understand how to get the coupled DEs. Presumably you mean changing variables or applying chain rule after reaching this second equation, still trying something out, but still lost nonetheless...
I'm just working on the...