A complex scalar field that describes a quantum mechanical system. The square of the modulus of the wave function gives the probability of the system to be found in a particular state.

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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 ...
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127 views

Product of position eigenvectors at different times

I've been thinking about this, and it might sound like a stupid question, but I can't seem to find an answer anywhere, here goes: Whenever we calculate expecation-values between two position ...
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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 ...
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343 views

Area under the graph of squared wave function

I was given a graph of square of the wave function of a hydrogen atom, against the distance of the electron from the nucleus (denoted by r). What I know is that the square of the wave function gives ...
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1answer
166 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 ...
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422 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 ...
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643 views

A few parity questions for simple harmonic oscillator

I think I understand that the solution to the Schrodinger equation for the SHO is based on the Hermite polynomials (and the Guassian function). The solution set of all even Hermite polynomials are a ...
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316 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 ...
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170 views

Possible Outcomes from Measuring a Hydrogen Atom

A hydrogen atom is characterized by the wavefunction $$\mid \psi \rangle =\sqrt{\frac{2}{7}}\mid 4\,2\,1\rangle +\sqrt{\frac{1}{7}}\mid 2 \,1\,\bar{1}\rangle+\sqrt{\frac{4}{7}}\mid 3\,2\,0\rangle$$ ...
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104 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 ...
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310 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 ...
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Understanding the Wave Function and Excited States

A wave function is an infinite dimensional vector space, how can it "live" in $\mathbb{R}^3$? Given the equation that is built like: $$\Psi (x,t) = \sum ^{\infty} _{n=1} c_n \psi _n (x) e^{-i E_n t / ...
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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 ...
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576 views

Wave function collapse

When we try to measure the position of a system, wavefunction collapses to form a spike. After a while, the wavefunction spreads again, and you take another measurement, the results will be different" ...
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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. ...
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269 views

Majorana wavefunction

I'm trying to compute the wavefunction for a Majorana state in an nanowire/superconductor hybrid system, like arXiv: Majorana Fermions and a Topological Phase Transition in ...
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396 views

Confusion about the probability cloud

What is the meaning of the electron probability cloud? I understood it to mean that the electron has a probability to be found in a certain postion before measurement, but now after reading ...
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Infinite Wells and Delta Functions

In considering a delta potential barrier in an infinite well, I can just enforce continuity at the potential barrier-it doesn't have to go to zero. Why then does it need to go to zero at the walls of ...
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739 views

Why is the wave function complex? [duplicate]

Why should an equation (TDSE) in which first time derivative is related to second space derivative have a solution that contains $i$?The wave function is supposed to be complex, but I am unable to ...
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4answers
443 views

Reconstruction of “wavefunction” phases from $|\psi(x)|$ and $|\tilde \psi(p)|$

Consider a "wavefunction" $\psi(x)$, which has a Fourier transform $\tilde \psi(p)$ Suppose that we know, for each $x$, $|\psi(x)|^2$, and that we know, for each $p$, $|\tilde \psi(p)|^2$. Have we ...
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197 views

In Schrödinger equation, can we get $\Psi$ if we know $\varphi(n)$?

Let $A$ be a mechanical quantity and we know its eigenfunction $\varphi(n)$. Can we get its wave function $\Psi$, if we measure $A$ for many (maybe infinite) times at the same time?
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Tunnelling through a Dirac potential barrier

I am reading a QM book by Griffiths, which says it is possible for wave particle to tunnel through a barrier formulated by a Dirac function. This function is known to peak at infinity and also ...
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156 views

Does Schrödinger equation have dual-property with Heat equation?

I have experimental data that Schödinger equation maintains high frequencies, while heat equation low. Does Schrödinger equation have some duality property with heat equation?
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263 views

How to understand wavefunction in quantum mechanics in math

I am reading some introduction on quantum mechanics. I don't understand all but I get the point that the wavefunction tells some probability aspects. In one book, they show one example of the ...
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How to find the wavefunction that solves an infinite square well with a delta function well in the middle?

Solutions for the wavefunction in an infinite square well with a delta function barrier in the middle are easily found online (see here for an example). I am wondering what the wavefunction is for an ...
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341 views

Quantum harmonic Oscillator analytic method

I'm using a book from Griffiths, I got really stuck about how he arrived at the approximate solution, is it just by trying( trial solution method?), I really appreciate any help on this. ...
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1answer
386 views

Given wave function at $t=0$, what is the process of deriving time dependent wave equation? [closed]

Suppose $$\Psi (x, t=0)=Ae^{i\alpha _1}\psi _1(x)+Be^{i\alpha _2}\psi_2(x)+Ce^{i\alpha _3}\psi_3(x).$$ If $\psi _n$ are the energy eigenfunctions how would I derive $\Psi (x,t)$? I am having trouble ...
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Overlap integral and probability

I have a question regarding how to extract probability from an overlap integral. Specifically, I am calculating the probability of a particle in a bound state in a delta potential $V=-\alpha ...
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Expectation value of total energy for the quantum harmonic oscillator [closed]

A particles unnormalized wavefunction is given as $$\psi(x)=2\psi_1+\psi_2+2\psi_3.$$ How can I find $\langle E\rangle $ without calculating $\langle T\rangle$ or $\langle V\rangle $ ...
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193 views

Workaround to fermion sign problem?

My (rather incomplete) understanding of the NP-hard fermion/numerical sign problem is that it occurs when attempting to converge on a wavefunction for many-body fermion systems (for example, a small ...
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153 views

Meaning of $C$ in wavefunction equation ($\Psi_{MO} = C_1\phi_A(1s) + C_2\phi_B(1s)$, where $C_1=\pm C_2$)

I've just cracked open a biophysics textbook and it's all fine up until the introduction of the letter C in a wavefunction equation, and declaring C1= ±C2 I've had lectures on eigenfunctions etc. ...
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Non-normalizable QM bound state in 4 spatial dimensions?

Edit 26/Sept/13: Fixed Typo in potential I'm solving the following (seemingly simple) quantum-mechanical problem in four spatial dimensions. In natural units ($\hbar^2/2m=1$), the Schrödinger ...
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157 views

Wavefunction's inner product

When two wavefunctions are orthogonal we can write that $$\langle\Psi_n|\Psi_m\rangle=\delta_{mn}$$ This means that $$\langle\Psi_1|\Psi_2\rangle=\langle\Psi_2|\Psi_1\rangle=0$$ But if the two ...
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478 views

Wave function decomposition

Problem: Given the wave function $\Psi_0=A\sin^2(\theta)$ along with the Hamiltonian operator of a physical system: $H=\frac{L^2}{2I}+g B L_z$, find the eigenvalues and eigenfunctions of $\hat{H}$ ...
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When Eigenfunctions/Wavefunctions are real?

When the Hamiltonian is Hermitian(i,e. beyond the effective mass approximation), generally under which conditions the eigenfunctions/wavefunctions are real? What happens in 1D case like the finite ...
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Why do wave packets spread out over time?

Why do wave functions spread out over time? Where in the math does quantum mechanics state this? As far as I've seen, the waves are not required to spread, and what does this mean if they do?
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Normalizing a Wave Function

How would I normalize the wave function: $ \psi (x)$ = $Ce^{i\rho_0x/\hbar}e^{-|x|/2\Delta x}$? I squared it, which got rid of the imaginary part-Then I considered breaking up the absolute value-but ...
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Normalizing Wave Functions

We normalize the wave function to $1$, but couldn't we also normalize it to $-i$ as $(-i)^2=1$? Does this not work? Is it equivalent?
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Quantum states as rays as opposed to vectors

I recently read that a quantum state is actually defined by a ray and not a vector. That is it is possible to multiply a state $\psi$ by any complex number $c\in \mathbb{C}$ and you won't be changing ...
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141 views

Explicit solutions to 2-d Dirac Equation

The 2-d Dirac equation without any constants is represented usually as $$i*dt (\phi) = D (\phi)$$ where $D = m\sigma_2-i\sigma_1dx-i\sigma_3dy$. Where can I find explicit closed form solutions to ...
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How to prove dp/dt = -dV/dx? Quantum mechanics [closed]

I got this problem from a book called Introduction to quantum mechanics, griffin 2nd edition. and I did not get why the solution says first term integrates to zero, integration by parts twice?! ...
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Implicit Postulate of Quantum Mechanics

Consider the following quantum system: a particle in a one dimensional box (= infinite potential well). The energy eigenstates wave functions all vanish outside the box. But the position eigenstates ...
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1answer
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A Simple Explanation for the Schrödinger Equation and Model of Atom? [closed]

I tried reading the Wikipedia article to no avail - I simply cannot understand the Schrödinger Equation (what does each of the variables mean, especially the wave function), and the Schrödinger Model ...
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2answers
160 views

Transition integral from 1-D cartesian into 3-D polar coordinate system

Lets say we have an electron which can be in two states. Its wavefunction for two states is then $\Psi=A\Psi_n + B\Psi_m$, where $\Psi$ is time dependent wave function. I know that the transition ...
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The probability of finding the electron in the H-atom

In the book Arthur Beiser - Concepts of modern physics [page 213] author separates the variables in the polar Schrödinger equation assuming: $$\psi_{nlm}=R(r)\Phi(\phi)\Theta(\theta)$$ then there a ...
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Confusion about wavefunction separability

A wavefunction is inherently a multi-particle function. If you have a container that is perfectly isolated from the external universe (not possible, but just imagine it) and filled with $n$ ...
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Wavefunction as a combination of two stationary states - how to find those states?

Lets say we have a particle in a infinite square well which has a wavefunction like this ($A$ is some constant and $d$ is the width of the well): \begin{align} A\left[ \sin \left(\frac{2 \pi ...
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247 views

Orbital of Hydrogen molecule

does anybody here know an analytical approximation of the bonding hydrogen orbital MOLECULE? I am looking for a good approximation to this orbital, that might be in some textbooks to get an ...
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The Energy Eigenvalue of a Wavefunction

I've been reading an introduction to quantum mechanics online, and while constructing the Schrodinger equation for a free particle, the equation $i\hbar \frac{d \Psi}{dt}=\hbar\omega\Psi$ is obtained. ...
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300 views

States and observables in quantum mechanics

I'm beginning learn quantum mechanics. As I understand, state is a map $\phi$ from $L^2(\mathbb R)$ such that $|\phi|^2$ describes probability density of a particle's position. By integrating ...