Homework Statement
The hamiltonian for a given interaction is
H=-\frac{\hbar \omega}{2} \hat{\sigma_y}
where
\sigma_y = \left( \begin{array}{cc} 0 & i \\ -i & 0 \end{array} \right)
the pauli Y matrix
Homework EquationsThe Attempt at a Solution
So from the time dependant schrodinger...
I'm also interested in this proof.
if i start out with
f(a)=\int_{-\infty}^{\infty}f(x)\delta(a-x) dx [1]
and make the change of variable x\rightarrow -t
\Rightarrow dx\rightarrow -dt
then
f(a)=-\int_{-\infty}^{\infty}f(-t)\delta(a+t)dt [2]
i'm a bit confused how...
Homework Statement
Evaluate the integral I_1 = \int_0^{2\pi} \frac{d\theta}{(5-3sin\theta)^2}
Homework Equations
The Attempt at a Solution
I start off by switching the sine term for a complex exponential e^{i\theta}=cos\theta +isin\theta
I will consider only the Imaginary...
Thanks for the replies. Is this because the function is even in the upper half of the complex plane?
I thought of doing this by integrating a contour in only the positive quadrent, ie:
(0,0) to (R,0)
(R,0) to (0,iR) along contour ω [a radial path of radius R from the real axis to the...
Homework Statement
evaluate the integral:
I_1 =\int_0^\infty \frac{dx}{x^2 + 1}
by integrating around a semicircle in the upper half of the complex plane.
Homework Equations
The Attempt at a Solution
first i exchange the real vaiable x with a complex variable z & factorize...
Sorry, I don't think this was very clear. I have done some more reading:
My likelihood function L(λ) is poissonian:
f(k;\lambda)=\frac{e^{-\lambda}\lambda^k}{k!}
Log Likelihood function is:
L(\lambda)=ln\left(\Pi_{i}^{n} f(k_i;\lambda)\right)
Heres where i get a bit lost, I think my...
Homework Statement
I have a set of data from the DAMA experiment in which a detector attempted to measure collisions with 'WIMP's [Weakly Interacting Massive Particles] as a candidate for dark matter. The detector records the time in days of a collision event. After binning the data and...
Homework Statement
an operator for a system is given by
\hat{H}_0 = \frac{\hbar \omega}{2}\left[\left|1\right\rangle\left\langle1\right| - \left|0\right\rangle\left\langle0\right|\right]
find the eigenvalues and eigenstates
Homework Equations
The Attempt at a Solution
so i...
ive just noticed i made a typo in the question
A = A_0e^{i\omega_r - ikz}
should read
A = A_0e^{i\omega_rt - ikz}
so there should be a time dependance.
sorry
x=x_0 e^{i\omega t - ikz} \hat{x}
is the general form of a plane wave, this would propagate in the z direction with constant amplitude. correct?
so initially the wavevector is complex, k = kr - i ki
when the wavevector is only real, such that k = kr
x_i = x_0e^{i\omega t - (ik_r + k_i)z}
x_f...
The simplest waveform i can think of that has constant amplitude is
x = x_0 cos(\omega t)
but wouldn't a decay indicate behaviour more like
x = x_0 e^{-\frac{t}{\tau}}cos(\omega t)
but then how does the wavenumber come into it?
Homework Statement
A wave is driven at z=0 with constant real frequency ωr propagates in the z direction, for z>0 the amplitude varies as:
A = A_0 e^{i\omega_r - ikz}
where k is complex
k=k_r - i k_i
if a wave with spatially constant amplitude and purely real wavenumber kr were...
If i were to make the substitution:
\phi = \frac{2\pi x}{L}
\frac{\phi}{x} = \frac{2\pi}{L}
does this imply that the limits of integration change from L to 2π ?
Homework Statement
An electron is confined to a 1 dimensional infinite well 0 \leq x \leq L
Use lowest order pertubation theory to determine the shift in the second level due to a pertubation V(x) = -V_0 \frac{x}{L} where Vo is small (0.1eV).
Homework Equations
[1]
E_n \approx...