## Well-Mix of States

Consider a particle of mass $m$ in the infinite square well of width $L$. Its initial wave function (at time $t=0$) is a coherent mixture of the second and third… Click here to read more

Physics tutorials, questions and answers

This site was started back in 2008, and features, among others, a collection of physics homework exercises and tutorials for all levels of physics students (high school and college).

We hope our site will help you advance in your studies and make it easier to understand physics, hence the world that surrounds us!

All the contents of this site were written by Valentin Bogatu (Physics, PhD).

Please help us spread the word. This site is free (supported by advertising) and your help is necessary to keep us going. Thank you for your visit.

Consider a particle of mass $m$ in the infinite square well of width $L$. Its initial wave function (at time $t=0$) is a coherent mixture of the second and third… Click here to read more

You are given a capacitor with the surface area $S$ having distance between the armatures $d$. The capacitor is charged with surface densities $\pm\sigma$ and the the voltage source is disconnected…. Click here to read more

What is the potential of the infinite slot from Griffiths 3.3, if you are given a varying potential $V(0,y)=V_0x/a$ as a boundary condition at $x=0$ For the given arrangement of… Click here to read more

For the spherically symmetric charge distribution located in free space, having $\rho=\left\{\begin{matrix}3r^2&\text{ for r<a}\\\rho=0&\text{ for r>a}\end{matrix}\right.$a) Use the potential definition and integrate the charge distribution to find the potential at… Click here to read more

For the asymmetric quantum well in the figure please find the first two bound states and energies. prove that it is possible for an asymmetrical well to have no bound… Click here to read more

Some researchers have been able to produce a wave packet that is somewhat similar to a “chopped wave plane”. Namely that a wave packet is just a slice from an… Click here to read more

The transmission probability for a particle of energy $E$ on an abrupt step of height $U_0$ when $E>U_0$ is $T=4k_1k_2/(k_1+k_2)^2$ where $k_1$ is the wave vector when $U=0$ and $k_2$… Click here to read more

Relativity 1. There is much talk today of relativity. The general public (which is not usually concerned with physical theories) become interested in this doctrine mainly because of its proposals… Click here to read more

1. For any two kets $|ψ>$ and $|χ>$ that have finite norm, show that$Tr(|ψ><χ|)=<χ|ψ>$ (Hint: chose matrix representation and use an arbitrary basis) $<χ|ψ>=(χ_1^*, χ_2^*, χ_3^*,….)\begin{pmatrix}ψ_1\\ψ_2\\ψ_3&…\end{pmatrix}=χ_1^* ψ_1+χ_2^* ψ_2+χ_3^* ψ_3+⋯$.$|χ><ψ|=\begin{pmatrix}χ_1^* ψ_1&χ_1^*… Click here to read more

Consider two operators represented in a three-dimensional space by matrices$H=\epsilon\begin{pmatrix}1&0&0\\0&-1&0\\0&0&-1\end{pmatrix}$ and $A=\alpha\begin{pmatrix}1&0&0\\0&0&1\\0&1&0\end{pmatrix}$ a) Are these operators Hermitian?b) Demonstrate that they commute.c) Find a basis of eigenvectors common to the two… Click here to read more

The NCC Civilization spaceship, having the length of 221 m is approaching the star gate SG-7 (having the same length 221 m) at the speed $0.9c$. A paint bomb on… Click here to read more

Please compute and choose the values of all components of the Audio Amplifier in the figure as to obtain a power of 6W. The computations and explanations for each component… Click here to read more

Demonstrate that whichever chosen the inertial system, a particle can not have a speed greater than the speed of light C. In other words show that there are no speeds… Click here to read more

You are given three charges two of them equal with $+q$ situated on the sphere of radius $a$ at $(theta,phi) =(pi/4, 0)$ and $(pi/4, pi)$ and the third equal with… Click here to read more

Please find for the Common Emitter, the values of $R_1$ and $R_2$ the voltage gain as and its input and output impedance. Values are $I_E=1.5 mA$ and $beta=100$. $I_c=beta I_b$… Click here to read more

You are given a particle that is free, having its state at $t=0$ equal that of the n-th energy level in the inifinite well.$\varphi=\sqrt{2/L}\sin (n\pi x/L)$ when $0<x<L$ a) Please… Click here to read more

A particle of mass $m$ and energy $E>0$ is incident from the left on the double delta potential $V(x)=V_0delta(x+a)-V_0delta(x-a)$ Find the wave function describing the stationary states of the particle as… Click here to read more

Recall the generalized uncertainty principle $\sigma_a \sigma_b \geq (1/2)<\psi|[A,B]|\psi>$. Note that the quantity $\sigma_a$ is known as the uncertainty in quantity $A$ (and similarly for $B$).a) Suppose you knew quantity… Click here to read more

Consider a system whose state vector $|\psi>$ and two observables $A$ and $B$ are given by$|\psi>=(1/\sqrt{5})\begin{pmatrix}-i\\2\\0\end{pmatrix}$; $A=\begin{pmatrix}1&i&1\\-i&0&0\\1&0&0\end{pmatrix}$; $B=\begin{pmatrix}3&0&0\\0&1&i\\0&-i&0\end{pmatrix}$ a) Are $A$ and $B$ compatible? Which set of operators ${A},{B},{A,B}$ form… Click here to read more

Consider a system with a hamiltonian $H=\frac{1}{\sqrt{2}}\begin{pmatrix}1&-i&0\\1&3&3\\0&3&0\end{pmatrix}$ in the initial state given by $|\psi_0>=\begin{pmatrix}4-1\\-2+5i\\3+2i\end{pmatrix}$ a) If the energy is measured, which values will be obtained, and with which probabilities?b) Find the… Click here to read more