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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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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0answers
63 views

Measurement and wavefunction collapse. problematic time in quantum mechanics

Q: When does the wavefunction collapse? A: When a measurement is made. But when exactly is this? I have a question about the time at which a measurement can be considered to have occurred: what ...
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4answers
336 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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1answer
192 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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1answer
302 views

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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2answers
149 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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207 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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1answer
2k views

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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1answer
227 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
252 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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1answer
954 views

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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1answer
602 views

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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2answers
139 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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1answer
136 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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3answers
141 views

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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1answer
107 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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326 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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3answers
583 views

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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1k views

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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1answer
125 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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1answer
952 views

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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3answers
229 views

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
358 views

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
144 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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3answers
282 views

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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2answers
708 views

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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1answer
202 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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1k views

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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3answers
240 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 ...
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443 views

Basic question on bra-ket notation

Which of the following corresponds to a $ \psi(x)$, a wavefunction written in the position basis: $ x| \psi\rangle $ or $ \langle x| \psi\rangle $? If it is the second expression (which my textbook ...
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233 views

What happens to the wave function after applying the D'Alembert operator?

Is the result of applying the D'Alembert operator on a wave function always zero? Or are there exceptions?
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1answer
793 views

Time Reversal Operator

I know that time reversal operator is an antiunitary operator. How does it work on wavefunctions? I believe in this way: $$T \psi (k,+)=e^{i\pi S_y/\hbar} K \psi (k,+) = \psi^*(-k,-),$$ but I am not ...
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1answer
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Variational Derivation of Schrodinger Equation

In reading Weinstock's Calculus of Variations, on pages 261 - 262 he explains how Schrodinger apparently first derived the Schrodinger equation from variational principles. Unfortunately I don't ...
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1answer
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Solution of 1-D Schrodinger equation for the potential $V(x) = -\frac{1}{|x|}$

May be this question might have already been asked but i couldn't find it, so let me know if its already there. Consider a potential, $V(x) = -\frac{1}{|x|}$ and if we apply this to a one dimensional ...
2
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1answer
690 views

Solving a time independent Schrödinger equation with a given potential

I'm trying to rework this old homework problem, and I am having problems arriving at the same solution on the answer sheet: Let $$V(x)=\begin{cases}\infty &\text{ if } x < 0\\ ...
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1answer
151 views

Wavefunction operators and the observable [closed]

So I got this from the exam I had yesterday. I couldn't really answer it other and it played on my mind through the night Show that if a wave function $\psi$ , is an eigenfunction of an operator [Q], ...
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3answers
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Writing wave functions with spin of a system of particles

Suppose I have 2 fermions in a potential $V(x)$. Both particles are moving in one dimension: the $x$ axis. Then, neglecting the interaction between the particles, the spatial wave function of the ...
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A quantum particle which is almost at rest but whose position is random!

Assume a particle is given by a quantum state which is constructed in such a way that it is equally probable to find it anywhere in an fixed interval $(0,L)$ but has arbitrarily low velocity. The ...
4
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2answers
386 views

Schrödinger equation in position representation

$$ \DeclareMathOperator{\dif}{d \!} \newcommand{\ramuno}{\mathrm{i}} \newcommand{\exponent}{\mathrm{e}} \newcommand{\ket}[1]{|{#1}\rangle} \newcommand{\bra}[1]{\langle{#1}|} ...
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3answers
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No well-defined frequency for a wave packet?

There are similar questions to mine on this site, but not quite what I am asking (I think). The de Broglie relations for energy and momentum $$ \lambda = \frac{h}{p}, \\ \nu = E/h .$$ equate a ...
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2answers
161 views

Decomposition of this wave function in eigenfunctions

I have this wave function of a system on a central potential: $V(r)$: $$\Phi(x,y,z)=C(x+y+z)e^{-\alpha r^2}.$$ And I'm asked a few things about probabilities. I don't have problems with that, because ...
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1answer
264 views

Problem from Sakurai about a delta-function potential [closed]

Can you help me with this problem from Sakurai: A particle of mass m in one dimension is bound to a fixed center by an attractive delta-function potential: $$V(x) ~= ~-a\delta(x) , \qquad ...
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1answer
838 views

Expectation value of momentum

I'm having a problem with an expectation value that doesn't seem to add up for me. What I know is, that $\psi(\vec{r})$ is a wavefunction for a particle in three dimensions. The Hamiltonian is given ...
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2answers
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Text interpretation in Griffith's intro to QM

It says in Griffith's chapter 2.1, that: $$\tag{2.14} \Psi(x,t)~=~\sum_{n=1}^{\infty}c_n\,\psi_n(x) e^{(-iE_n t/\hbar)}$$ It so happens that every solution to the (time-dependent) Schrodinger ...
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Expectation value of position in infinite square well

I'm looking for some help to a question. I'm working in the infinite square well, and I have the wavefunction: $$\psi(x,t=0)=A\left( i\sqrt{2}\phi_{1}+\sqrt{3}\phi_{2} \right).$$ For every time t, ...
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First-order pertubation theory

I'm having some trouble figuring this out, so I was hoping someone could help. I need to show that the first-order pertubation of the ground state energy is not changed by the pertubation $H'$, given ...
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1answer
283 views

Infinite Potential Well Energy for Piece-wise Constant Wave Function

I'm trying to compute the expectation value of energy for a certain state in an infinite potential well but I'm getting contradictory answers. The well has potential \begin{align} V(x) = \left\{ ...