For part c) the integral is performed under all space.
When you write out the integral, the integrand is [sin(\frac{x}{3})]^*cos(\frac{x}{3}) with a constant on the outside of the integral. Now for this to be 0. X would have to be those values that I wrote in my previous post. Does this seem...
Homework Statement
a) Show that the functions f=sin(ax) and g=cos(ax) are eigenfunctions of the operator \hat{A}=\frac{d^2}{dx^2}.
b) What are their corresponding eigenvalues?
c)For what values of a are these two eigenfunctions orthogonal?
d) For a=\frac{1}{3} construct a linear...
Homework Statement
Prove that if two linear operators A and B commute and have non-degenerate eigenvalues then the two operators have common eigenfunctions.
Homework Equations
[A,B]= AB - BA= 0
Af=af
Bg=cg,\ let\ g=(f+1) --> B(f+1)=c(f+1)\ where\ a\neq c
The Attempt at a...
Solve the following differential:
\frac{d^2}{d\theta^2} + cot\theta\frac{dS}{d\theta} - \frac{m^2}{sin^2\theta}S(\theta) + \frac{cS(\theta)}{\hbar}=0
The first step is:
"For convenience we change the independent variable, by making the substitution w=cos\theta"
So my question is...
What does this mean?? (Intro QM)
I am on the angular momentum unit in my introductory quantum mechanics course and we had to examine the following two commutators:
1) [Lx,Ly] it ends up equating to i\hbar Lz
[Lx,Ly]= i\hbar Lz
[Ly,Lz]= i\hbar Lx
[Lz,Lx]= i\hbar Ly
Tt was concluded...
I don't believe there are any specific parentheses. The exact question is to find the following commutator [Lx,Ly]
where:
Lx= (y\frac{\partial}{\partial z} - z\frac{\partial}{\partial y})
Ly= (z\frac{\partial}{\partial x} - x\frac{\partial}{\partial z})
Homework Statement
Simplify the following two expressions:
y\frac{\partial}{\partial z}z\frac{\partial}{\partial x}
z\frac{\partial}{\partial y}x\frac{\partial}{\partial z}
The Attempt at a Solution
for the first one: y\frac{\partial}{\partial z}z\frac{\partial}{\partial x}...
The Right Hand Side of the equation gives me the difficulty. I am sure I am making an elementary mistake.
\frac{\partial\theta}{\partial y}:
\frac{d}{dy}cos\theta=\frac{d}{dy}(\frac{z}{r})
\frac{d}{d\theta}(cos\theta)\frac{d\theta}{dy}=\frac{d}{dr}r^-1\frac{dr}{dy}...
An earlier part to this question was to find \frac{\partial r}{\partial y} and I solved it correctly.
Here is how I did it.
r^2= x^2 + y^2 + z^2
\frac{d}{dy}r^2= \frac{d}{dy}y^2
\frac{d}{dr}r^2\frac{dr}{dy}=2y
2r\frac{dr}{dy}=2y
so therefore \frac{\partial r}{\partial y}=...
Homework Statement
I am supposed to find if the following commutes: [Lx,Ly]
Homework Equations
Lx= -i\hbar[y(\partial/\partialz) - z(\partial/\partialy)]
Ly= -i\hbar[z(\partial/\partialx) - x(\partial/\partialz)]
where [Lx,Ly]=LxLy-LyLx
If it commutes then [Lx,Ly]=0...