1
vote
1answer
107 views

Doubt in a certain equation of a research paper [closed]

In the given paper, I am stuck at equation (7). The equation that I am trying to solve for particle outside the well is : (1/g)*(g'') + (1/(r*g))*g' - (k_o)^2 = 0 where g = Radial wave function. r = ...
0
votes
1answer
38 views

Expected value $<\hat{x}>$ of: $\Phi(x,t)=Ne^{-a[(Mx^2/\hbar)+it]}$ is infinite, why?

The problem says: A particle of mass $M$ is described by the wave function: $$\Phi(x,t)=Ne^{-a[(Mx^2/\hbar)+it]}$$ where a is a positive constant. Asked to determine such things as the ...
0
votes
0answers
65 views

Complex Conjugate of Wave Function's Derivative

I am reading Griffiths QM textbook and I got confused by the following identity: How to prove from $$\frac{\partial \Psi}{\partial t} = \frac{i\hbar}{2m} \frac{\partial^2 \Psi}{\partial x^2} - ...
1
vote
2answers
105 views

Determining the Wave Function From Initial Conditions

This is Problem 2.6 (b) in Griffiths, Intro to QM: A particle in an infinite square well has its initial wave function an even mixture of the first two stationary states: $\Psi(x,0) = ...
0
votes
1answer
73 views

Determine the normalisation constant of a piecewise wavefunction

I'm trying to find the normalisation constant $N$ for the following wavefunction: $$ \psi\left(x\right) = \left\{ \begin{array}{lr} N \left(x^2 - l^2\right)^2 &\: \left|x\right| \le l ...
0
votes
0answers
51 views

Finite Square Well Inside an Infinite Square Well

Ok here's a potential I invented and am trying to solve: $$ V(x) = \begin{cases} -V_0&0<x<b \\ 0&b<x<a \\ \infty&x>a \\ \end{cases}$$ and $V(-x) = V(x)$ (Even ...
3
votes
1answer
42 views

Probability of measuring momentum [closed]

Suppose we have this wavefunction: $$ \psi = A \left( cos(kx) + cos (2kx) \right) $$ I have to find the possible results of measurement of momentum and their probabilities. Attempt For a momentum ...
1
vote
1answer
71 views

Obtain the eigenfunction of Jz for the wave function of an electron in a hydrogen atom? [closed]

The wave function of an electron in a hydrogen atom is given by Is this wave function an eigenfunction of Jz , the z-component of the electron’s total angular momentum? If yes, find the ...
1
vote
1answer
100 views

Why don't we need to normalize wavefunction to find probability distribution?

Consider an unormalized wavefunction of a rotor at $t = 0$, a combination of $n=0$ and $n=2$ states: $$\psi(\phi) = 3 - 2 \cos (2\phi).$$ Find the probability distribution in angle. The ...
0
votes
1answer
195 views

Calculating the expectation value of a Hamiltonian

I want to calculate the expectation value of a Hamiltonian. I have a wave function that is $$\psi = \frac{1}{\sqrt{5}}(1\phi_1 + 2\phi_2).$$ I want to know if I set this up properly. The ...
0
votes
1answer
99 views

Normalizing the sum of wavefunctions and calculating probabilty - understanding concepts

A state of a particle bounded by infinite potential walls at x=0 and x=L is described by a wave function $\psi = a\phi_1 + b\phi_2 $ where $\phi_i$ are the stationary states. So let's say we want to ...
1
vote
1answer
105 views

Normalisation of Linear Harmonic Oscillator - Ladder Operator Method

I was watching the following video on the harmonic oscillator using ladder operators : http://youtu.be/gRdCV9p8sAU?t=30m9s Clicking on the video above will take you to the exact point where my ...
1
vote
2answers
102 views

Quantum Mechanics: Momentum operator questions [closed]

I'm asked to determine $\hat{P}|\Psi_0\rangle$, $\langle{\hat{P}}\rangle$, and $\langle\hat{P}^2\rangle$ for $$\Psi_0(u) = \psi_0 + 2\psi_1$$ I understand how to make the matrix for $P$ in regards ...
0
votes
2answers
41 views

Calculating the Probability Current of a Travelling Wave

Calculate the probability current density vector $\vec{j}$ for the wave function : $$\psi = Ae^{-i(wt-kx)}.$$ From my very poor and beginner's understanding of probability density current it is : ...
4
votes
1answer
104 views

Orthogonality of summed wave functions

Problem. I know that the two wave functions $\Psi_1$ and $\Psi_2$ are all normalized and orthogonal. I now want to prove that this implies that $\Psi_3=\Psi_1+\Psi_2$ is orthogonal to ...
-1
votes
1answer
105 views

Apply the Heisenberg Equation to the Hamiltonian [closed]

$\frac{d}{dt}$$\hat{H}$ = $\frac{i}{\hbar}$$[\hat{H},\hat{H}]$ +$\frac{\partial{\hat{H}}}{\partial{t}}$ That's as far as I've got. I do not know much about the Heisenberg equation or even what it ...
0
votes
1answer
52 views

Wave function - Working towards Orthonormal solution

As you can see below, the two pages attached describe how to obtain the final solution of orthonormality. I do not completely follow the discussion but I will talk you through what I DO understand ...
0
votes
1answer
56 views
0
votes
0answers
91 views

Difference between expectation values of $L^2$, $L_z$ and measuring $L^2$, $L_z$

I was given with this hydrogen radial wavefunction $$ R_{21} =\left(\sqrt{\frac{1}{3}}Y^0_1 + \sqrt{\frac{2}{3}}Y^1_1\right) $$ and was asked to find a) What are the expectation values of the ...
0
votes
1answer
158 views

Expectation value of energy from the position state of hydrogen atom [closed]

I was given with the position state of hydrogen atom: $$ R_{21} =\left(\sqrt{\frac{1}{3}}Y^0_1 + \sqrt{\frac{2}{3}}Y^1_1\right) $$ I am getting confused about getting the expectation value of ...
3
votes
1answer
186 views

Harmonic Oscillator potential, proof that Gaussians remain Gaussians?

I read in several papers that for a Harmonic Oscillator Hamiltonian in the time dependent Schrödinger equation a Gaussian wave packet remains Gaussian. Unfortunately I could not find any proof for ...
2
votes
1answer
147 views

Two ways of calculating the expectation value of momentum

The expectation value of momentum is given by: $$ \langle p\rangle = \int_{-\infty}^{\infty}\psi^{*}(x)\left(-i\hbar\frac{\partial}{\partial x}\right)\psi(x)dx $$ How can I show that the above ...
0
votes
1answer
66 views

Group velocity of localized wavepacket

This is a Homework problem so please feel free to not answer and just give pointers. A localized wavepacket is given as: $$\Phi(r,t=0 ) = \frac { e^{-\large\frac {r^2}{2s^2}} e^{\large\frac{i\pi ...
0
votes
2answers
347 views

Schrödinger's Equation and the depth of a finite potential well

Before I ask my question, I have to stress: I have absolutely no idea what the math is going on. I've read my textbook, several Wikipedia articles, scoured the internet, and don't feel anymore ...
1
vote
1answer
157 views

Energy difference between symmetric and antisymmetric wavefunctions

Is there any energy difference between a particle in a symmetric wavefunction and an identical particle in an identical potential but in a state with an anti-symmetric wavefunction? Or is it ...
5
votes
2answers
94 views

Physical position eigenfunction normalisation

We know that the Dirac function $$\delta(a)=\lim_{a \rightarrow 0} \delta_{a}(x)$$ can be written as an infinitesimally narrow Gaussian: $$ \delta_{a}(x) := \frac{1}{\sqrt{2\pi a^2}}e^{-x^2/2a^2}$$ ...
0
votes
1answer
128 views

Wavefunction Problem wrong in solutions manual? [closed]

Well there is a problem in my book which lists this problem: Calculate the probability that a particle will be found at $0.49L$ and $0.51L$ in a box of length $L$ when it has (a) $n = 1$. Take the ...
2
votes
1answer
110 views

Three dimensional wave packets in momentum space

I am given the 3D wave packet: $$\psi(x,y,z)=N\,\exp\left(\frac{-(x^2+y^2+2z^2)}{2a^2}\right).$$ I was asked to find N (easy enough). Then I was asked the probability that we measure $z$ greater than ...
2
votes
0answers
89 views

Momentum representation of a state

I am trying to figure out the momentum representation of the state which has the properties $$\langle \psi |\hat q |\psi \rangle=-q_0,$$ $$\langle\psi|\hat p|\psi \rangle=p_0, $$$$\Delta q\Delta ...
2
votes
1answer
67 views

Transition from coordinate space to momentum space for SHO

I am given that the ground state of the SHO in position space is given as $$\langle q|\psi_0\rangle=\frac{1}{a^{\frac12}\pi^{\frac14}}e^{-q/4a^2}$$ Where a is a constant with units of length. I am ...
1
vote
1answer
137 views

photon polarization, uncertainty in Energy

A beam of red light is sent along the $z$ axis through a polaroid filter that passes only $x$ polarized light. The beam is initially polarized at $30$°, and the total energy is $10$ Joules. Estimate ...
2
votes
2answers
112 views

Infinite well particle subject to additional time dep. potential

I am asked to find the wavefunction of the particle in a well subject to an additional potential $$V(x,t)=\frac{\pi x \hbar}{L}\delta(t).$$ I have already solved that ...
0
votes
1answer
70 views

Question on measuring expectation value of spin with time variation

I have a particle with the following wave function: $$\psi(t) = \frac12 |\uparrow \rangle e^{-i(\omega_1+\omega_2)t/\hbar} +\frac12 |\uparrow \rangle e^{-i(\omega_1-\omega_2)t/\hbar} ...
1
vote
0answers
119 views

Basic introductory quantum mechanics question [closed]

Given that $\psi = \frac{1}{\sqrt{32 \pi a_{0}^{3}}}(2-\frac{r}{a_0})exp(\frac{-r}{2a_0})$ is a wavefunction of the hydrogen atom, write down the probability density for r and calculate the ratio ...
2
votes
2answers
300 views

Solving quantum radial equation for infinite potential spherical annulus for $l=0$

There is a mass $m$ in a potential such that $$ V(r) = \left\{ \begin{array}{lr} 0, & a \leq r \leq b\\ \infty, & \text{everywhere else} \end{array} \right. $$ ...
4
votes
1answer
170 views

Virial theorem and variational method: an exercise (re-edited)

I have a hydrogen atom, knowing that its Hamiltonian has been modified turning the standard potential $$ V_{0}(r) = -\frac{Z}{r} $$ into $$ V(r) = -\frac{g}{r^{\frac{3}{2}}} $$ with $g$ a positive ...
1
vote
1answer
148 views

Spin eigenvalues and eigenvectors problem. Is this the correct way to solve it?

An electron is described by the Hamiltonian $ H=\frac{e}{mc}\bar{S}\cdot\bar{B} $ where $\bar{S} =(S_x,S_y,S_z)$ is the spin operator and $\bar{B}$ the magnetic field. For $t>0$ ...
1
vote
1answer
193 views

Energy and time evolution of a particle in a potential well

Hoping this is not a silly and stupid question let me ask for help in this problem. I have a particle in an infinite square well (the box is from 0 to a), in the state described by the function ...
7
votes
1answer
264 views

Virial theorem and variational method: a question

I have an hydrogenic atom, knowing that its ground-state wavefunction has the standard form $$ \psi = A e^{-\beta r} $$ with $A = \frac{\beta^3}{\pi}$, I have to find the best value for $\beta$ ...
1
vote
2answers
194 views

Bloch wave function orthonormality?

there is this text book that is giving me a hard time for a while now: It shows that Bloch wave functions can be written as $$\Psi_{n\vec{k}}\left(\vec{r}\right) = \frac{1}{\sqrt{V}}e^{i\vec k \vec ...
1
vote
1answer
1k views

Ground State Wavefunction of Two Particles in a Harmonic Oscillator Potential

Question: Two identical, non-interacting spin-$1/2$ particles are in a 1D Harmonic Oscillator Potential. Their Hamiltonian is given by ...
-3
votes
2answers
64 views

Which number should I suppose to $a$ (width of well) and $m$ (mass of particle) in potential well problem? [closed]

I tried to plot a complete of state functions of potential well problem but graph was so weird. I thought a cause was variables a and ...
1
vote
1answer
141 views

Modulus Square of the Gaussian Wave Packet for uncertainty in $p$

Upon evaluating the integral (2.67) and obtaining the complex valued equation given in box 2.4, the author performs the modulus square to obtain the Gaussian distribution (2.68). How does one go about ...
0
votes
2answers
294 views

Normalising a wavefunction where $\psi$ is equal to a sum of functions [closed]

The wavefunction $\psi(x)$ = $\phi_1(x)$ + $2\phi_2(x)$ + $3\phi_3(x)$ is to be normalised. The functions $\phi_1(x)$, $\phi_2(x)$, $\phi_3(x)$ are normalised eigenfunctions of a Hermitian operator ...
3
votes
2answers
289 views

Formulation and probability of a wave-function [closed]

I have got this problem where I have been given the following wave function: $$\Psi = 0\quad\text{if}~|x| > a\quad\text{and}\quad A(a^2-x^2)\quad \text{if} \quad |x|< a$$ Now the first question ...
2
votes
1answer
135 views

Possible Outcomes from Measuring a Hydrogen Atom

A hydrogen atom is characterized by the wavefunction $$\mid \psi \rangle =\sqrt{\frac{2}{7}}\mid 4\,2\,1\rangle +\sqrt{\frac{1}{7}}\mid 2 \,1\,\bar{1}\rangle+\sqrt{\frac{4}{7}}\mid 3\,2\,0\rangle$$ ...
0
votes
1answer
93 views

Momentum representation of a function with discontinuous derivative [closed]

Consider the following wave packet $$\psi = Ce^{2\pi i p_0x/h}e^{-|x|/(2\Delta x)}$$ where $h$ is the Planck's constant and $C$ is the normalization constant. The derivative of this function is ...
2
votes
1answer
234 views

Probability current vs. direction of wave function

I did an exercise for my Quantum-Mechanics Lecture: Let $\hbar$=2m=1. A particle in 1 dimension has $j(x)=2\ Im(\overline{\psi} (x) \ \psi'(x))$ and it's to show that there are superpositions $\psi ...
5
votes
1answer
497 views

Orthonormality of Radial Wave Function

Is the radial component $R_{n\ell}$ of the hydrogen wavefunction orthonormal? Doing out one of the integrals, I find that $$\int_0^{\infty} R_{10}R_{21}~r^2dr ~\neq~0$$ However, the link below says ...
1
vote
1answer
552 views

Writing a Wavefunction as a Linear Combination of Eigenstates

We have the following wavefunction for the hydrogen atom: $$\psi(r,\theta,\phi)=\frac{1}{\sqrt{4\pi}}\frac{1}{(2a)^{3/2}}\frac{r}{a}e^{-r/2a}\sin(\theta)\sin(\phi)$$ where $a$ is the Bohr radius. ...