Homework Statement
The potential of a simple harmonic oscillator of HF has the following form
\frac{1}{2}kx^2 + bx^3 + cx^4
The first part of the problem involved finding expressions for the first-order energy corrections for the first three states, which I found below. Basically the x3 term...
Homework Statement
Prove that a continuous linear functional, f is bounded and vice versa.
Homework Equations
I know that the definition of a linear functional is:
f( \alpha|x> + \beta|y>) = \alpha f(|x> ) + \beta f( |y> )
and that a bounded linear functional satisfies:
||f(|x>)) ||...
Homework Statement
Is the following an inner product space if the functions are real and their derivatives are continuous:
<f(t)|g(t)> = \int_0^1 f'(t)g'(t) + f(0)g(0)
Homework Equations
I was able to prove that it does satisfy the first 3 conditions of linearity and that...
Hi all,
I was diving into my 3rd year quantum assignment and I saw the following which I have to use for the rest of the question to prove the Cauchy-Schwarz inequality:
Homework Statement
|| a|x> + b|y> ||^2
I only really learned a bit about Dirac notation last year, so please...
Homework Statement
u_{xxx} - 3u_{xxy} + 4u_{yyy} = e^{x+2y}
The Attempt at a Solution
Ok so I tried doing the following to solve the homogeneous equation
\begin{align*}
u_{xxx} + u_{xxy} - 4u_{xxy} + 4_{yyy} = 0 \\
[d^{2}x(dx -dy) - 4dy(dx + dy)(dx -dy)]u = 0 \\
[(dx -...
Hi all,
Hopefully this is the right section for my post, if not I apologize.
I'm hoping I can just get some advice to help me get started in the right direction. I am trying to do a mathematical inversion for the following:
\frac{1}{N(zi)} \frac{dN}{dz}|_{z=zi} = -\frac{2}{zi} -...
Homework Statement
There is an infinite slab of material with magnetic susceptibility \chi_m parallel to xy plane and between z=-a and z=a. There is a free current with density J=J_0 \frac{z}{a} in the x direction, so it's positive for z>0, and negative for z<0.
What is the H field...
Homework Statement
There is an infinite slab of material with magnetic susceptibility \chi_m parallel to xy plane and between z=-a and z=a. There is a free current with density J=J_0 \frac{z}{a} in the x direction, so it's positive for z>0, and negative for z<0.
What is the H field inside...
2 Masses 1 Spring Question! Help
Homework Statement
Two masses m1 and m2 slide freely on a frictionless horizontal plane, and are
connected by a spring of force constant k . Find the natural frequency of oscillation for
this system.Homework Equations
\ddot{x} + \omega^2x=0 where \omega^2...
Homework Statement
Calculate the energy per unit length for two long coaxial cylindrical shells, neglecting end effects. The inner and outer cylinders have radii a and b, and linear charge densities λ and -λ, uniformly distributed on the surface, respectively.
2. The attempt at a solution...
I don't think so since the prof asked to solve it in terms of t explicitly. Maybe she made a mistake in writing the problem if this cannot be solved analytically.
Ok well it was a differential equation problem that I reduced to that but here is the initial problem:
dy/dt + y\cos(t) = 1/ (1+t^2)
so I got an integrating factor of e^(sint) which led to this integral! Hope this helps maybe I did something wrong in first part.
Homework Statement
Integrate: \exp(\sin(t)) / (1 + t^2)
The attempt at a solution
Ok so I tried substituting u=sin(t) du=cos(t)dt but I end up with (1 + arcsin^2(u)) on the bottom and I don't know how to integrate that.
I also tried letting t=cos(u) dt=-sin(u)du but then I end up...