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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489 views

Bound states of the $V(x)=\pm \delta'^{(n)}(x)$ potential?

The $\delta(x)$ Dirac delta is not the only "point-supported" potential that we can integrate; in principle all their derivatives $\delta', \delta'', ...$ exist also, do they? If yes, can we look for ...
0
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1answer
48 views

Fourier expansion and transform - what about the phase of the waves that i am adding?

Say we have a wave on the surface of the water and we want to describe it as a sum of other waves. So we use Fourier expansion to add waves of different wavelengths. For simplicity, say we have to ...
4
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2answers
560 views

Schroedinger equation for hydrogen atom

I have got a problem understanding the meaning of the Laplace operator in the Schrödinger equation for the hydrogen atom. $$\Big(-\frac{\hbar^2}{2m_e} \Delta_{r_e} - \frac{\hbar^2}{2M_P} \Delta_{r_p} ...
1
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1answer
798 views

What does a light wave look like (3d model)

What does a light wave look like? The only models I can seem to find online are 2D waves, they just look like sin() graphs. I have seen the models of the two components of "light waves" (electric ...
2
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4answers
10k views

Differences between wavefunction, probability and probability density?

I am trying to understand the differences between wavefunction, probability and probability density. There are different definitions on the internet. For example: http://inside.mines.edu/~fsarazin/...
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4answers
417 views

Physical intuition behind negative values for wave function?

So a positive and a positive wave function create a bonding orbital where the probability of finding an electron is summed while a positive and a negative create an anti-bonding orbital with a lower ...
8
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1answer
1k views

Must the derivative of the wave function at infinity be zero?

I came across a problem in Griffiths where the derivative of the wave function (with respect to position in one dimension) evaluated at $\pm\infty$ is zero. Why is this? Is it true for any function ...
1
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1answer
144 views

Is the photon's wave function the same as an electromagnetic wave(light)? [duplicate]

The first that i have been taught in Quantum Mechanics is the photoelectric phenomenon. Without analyzing it, it concludes that when we shine light at the circuit(roughly speaking), the work required ...
4
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2answers
111 views

Do quantum wave functions curve spacetime before they are measured

Do wave functions cause spacetime curvature before they are measured, or would curvature only happen upon measurement? I guess the question becomes, do quantum wavefunctions carry energy while they ...
1
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1answer
245 views

Width of a 1 dimensional box with same ground state energy as hydrogen atom [closed]

I am trying to find the width $L$ of a one-dimensional box for which the ground state energy of an electron in the box equals the absolute value of the ground state of a hydrogen atom. No ...
0
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2answers
329 views

Why is $ \psi = A \cos(kx) $ not an acceptable wave function for a particle in a box?

Why is $ \psi = A \cos(kx) $ not an acceptable wave function for a particle in a box with rigid walls at $x=0$ and $x=L$ where $$ k = \frac {(2mE)^{1/2}} {\hbar} \, ?$$ I had plugged the wave ...
0
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2answers
91 views

How can a solution of the time-independent Schrödinger equation evolve in space?

I understand that if the Hamiltonian does not depend on the time, the Schrödinger Equation becomes separable, so you get $$ H \psi(x) = E \psi(x) $$ and $$ \Psi(x,t) = \psi(x)\exp\left(-\frac{\...
1
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1answer
51 views

Potential energy function for high energy continuum?

For the hydrogen atom the quantised energy levels are: $$E_n = \frac{- 13.6 eV}{n^2}\quad\text{with}\quad n = 1,2,3...$$ One peculiar property of this quantisation is that for large $n$ the energy ...
0
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2answers
1k views

Normalization of Momentum Eigenfunctions: the number of particles

After finding the eigenfunctions $u_p(x)=Ce^{ipx/\hbar}$ of the momentum operator just like in this UCSD lecture notes, one seeks to normalize them, so one first tries: $$\int\limits_{-\infty}^{\infty}...
5
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3answers
371 views

Confusion about 1-forms as introduced in “Gravitation” (Kip S. Thorne,…)

In the book Gravitation in chapter 2, paragraph 5, they introduce the concept of 1-forms by thinking about the momentum 4-vector differently. They first introduce the de Broglie 1-form as follows (I ...
1
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2answers
257 views

How to predict bound states in a 1 D triangular well?

Assume we have a (single) particle in a potential well of the following shape: For $x \leq 0$, $V = \infty$ (Region I) For $x \geq L$, $V = 0$ (Region III) For the interval $x > 0$ to $x < L$,...
0
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1answer
149 views

Schrodinger's equation with negative sign

In time dependent Schrodinger's equation as given in Schrodinger's lecture (Four Lectures on Wave mechanics, Blackie & Son, 1949, pg22) he arrives at $$\nabla^2\psi-\frac{4 \pi m i}{h}\dot\psi-\...
9
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2answers
3k views

What is probability current in quantum mechanics?

What is probability current in quantum mechanics? Why define such a thing? I mean the meaning of probability current. I know the formula for it but I just don't get the idea of a flow of probability ...
2
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1answer
69 views

Normalizing a wave function in a mixed well

So I got this potential and want to solve for the even wavefunctions http://imgur.com/GKAy4nD Since it's symmetric around the origin I need only to look at the interval [0,b] and solve for the ...
-1
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2answers
355 views

What do “ℜe” and “A*” mean?

What do "$\mathfrak{Re}$" and "A*" mean in the following equation (taken from James Binney and David Skinner's QM lecture notes, equation 1.12), \begin{align} p(S\text{ or }T)&=\left|A\left(S\text{...
2
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1answer
52 views

Why do the two amplitudes need to match together through the region between the boxes?

This is an excerpt from Feynman's lectures 3; Suppose we think of the situation in Fig. 7–3, which has two boxes held at the constant potentials $ϕ_1$ and $ϕ_2$ and a region in between where ...
6
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1answer
1k views

Why does the wave function have to be continuous? [duplicate]

When solving one dimensional problems in quantum mechanics it is often assumed that the first derivative of the wave function is continuous. What justifies this assumption?
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2answers
120 views

Why can the probability function for a particle in an infinite square well be larger than 1?

For a particle in a one dimensional infinite potential well of width $L$ the probability function is: $$P_n(x)=\left(\frac{2}{L}\right)\sin^2\left(\frac{n\pi x}{L}\right)$$ for $0\leq x\leq L$. The ...
1
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3answers
1k views

Electron as a standing wave and its stability

1. When it was an era of classical mechanics we used to believe in the Bohr's atomic model. It interpreted electrons as particles (although I couldn't understand how come Bohr who interpreted ...
0
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3answers
578 views

Double slit experiment observation

In the double-slit experiment, if you shoot particles through the slits one by one and observe which slit they travel through, is there still an interference pattern on the screen behind the slits? If ...
2
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3answers
213 views

Wave Function concept

What do we mean when we say wave function of electron? Does it mean wave nature of electrons? I am really confused.Without clearing this confusion i cannot proceed to molecular orbital theory.I am ...
4
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4answers
199 views

Isn't the 'slit' in a double-slit experiment also a wave?

I'm new to QM so excuse my naivety. I was watching an online MIT QM course that described the double-slit experiment (with electrons) when it occurred to me that I have a question. In the video, the ...
1
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2answers
87 views

Where does the factor of $x$ come from in this formula for expectation value?

Given the normalised ground-state wave-function: $$\Psi(x, t)=\begin{cases} \sqrt\frac{2}{d}\cos(\frac{\pi x}{d})e^\frac{-i\hbar\pi^2t}{2md^2} & \ \lvert x\rvert<\frac{d}{2}, \\ 0 & \text{...
0
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0answers
24 views

Cancelling waves and preservation of energy

In quantum physics, a particle is "defined" by a wavefunction. If you would take 2 particles with the same wavefunction, and negate one of them. They would cancel each other other out. Take for ...
0
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0answers
28 views

Solutions to time-independent Schrödinger's equation with symmetrical (even) potential [duplicate]

A problem from Griffith's Introduction to Quantum Mechanics asks to prove the following: Given a symmetric potential $V(x)$ $(=V(-x))$, the solutions to the time-independent Schrödinger's equation ...
6
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2answers
202 views

What state the wave function collapses into after an inaccurate measurement?

I'm watching MIT online lectures Quantum Physics I (roughly from one hour mark in the video). The lecturer explains wave functions that describe "Stationary States" that consist of a single energy ...
2
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1answer
162 views

The definition of the vacuum state of quantum field by path-integral

In the review Entanglement entropy of black holes by Sergey Solodukhin (arXiv:1104.3712, equation 13), I see a definition of vacuum state of quantum field by path integral over half of the total ...
1
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1answer
39 views

$\sqrt{\frac{\omega ^2}{c^2}-k_z^2}$ in cylindrical harmonics

The radial component of the solution of the wave equation in cylindrical coordinates is $$J_\nu \bigg(\rho\sqrt{\frac{\omega ^2}{c^2}-k_z^2}\,\,\bigg).$$ But I always thought that $\frac \omega c$ ...
1
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0answers
66 views

Transfer function of a space varying wave equation

$$\frac{\partial ^2 \psi}{\partial x^2}-\mu \epsilon \frac{\partial ^2 \psi}{\partial t^2}-\mu \sigma \frac{\partial \psi}{\partial t}=0$$ Is the wave electromagnetic wave equation in lossy, source ...
3
votes
1answer
220 views

Plane wave expansion of cylindrical functions:Summation of the Hankel functions

I understand that; in cylindrical coordinates, the basic solutions of the Helmholtz equation are of the form Hankel function of integer order times a complex exponential term ($E=H_{n}(kr)e^{in\alpha}...
3
votes
3answers
317 views

Constructing solutions to the time-dependent Schrödinger's equation

The following question is from David Griffiths' Introduction to Quantum Mechanics: Problem 2.13 A particle in the harmonic oscillator potential starts out in the state $$\Psi(x,0) = A[3 \psi_0(x)...
0
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2answers
76 views

Does a free electron, one that's not either in an atom or a wire, have an associated wave-function?

Would a free electron, one that's not either in an atom or moving through a wire, but moving through empty space on its own, have an associated wave-function? Or, is an electron described as a wave-...
1
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1answer
97 views

Solution of the Radial Part of the Schroedinger Equation [closed]

The general Schroedinger Equation is: $$\left[-\frac{\hbar^2}{2m}\triangle +V(r,\vartheta,\varphi)\right]\psi_{nlm}=E\psi_{nlm}$$ When considering free waves, i.e. $V(r,\vartheta,\varphi)=0$ and a ...
0
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1answer
63 views

Property of the wave functions of a free particle

How can I show that the following holds? $$\langle nlm\mid \partial_z^2\mid nlm\rangle=-\int_0^{4\pi}d\Omega\int_0^{\infty}drr^2\left|\partial_z\psi_{nlm}\right|^2$$ The wave functions of a free ...
1
vote
2answers
122 views

How do phase carries structural information about the function? [closed]

Suppose you are on a railway platform and you hear the sound of train coming towards you. Now, Using Fourier transformation we can convert the time domain function (here take sound as a function) ...
3
votes
2answers
144 views

Time evolution of a wavepacket

I do not understand why if $H\psi = E\psi$, then the time-evolution of the wavefunction is given by $e^{-iEt/h}\psi(x)$.
0
votes
1answer
58 views

For an event that can occur in many ways, why is the wavefunction of the event the sum of wavevfunction for each way separately?

The wavefunction of identical particles is given as: $$\psi_{1,2} (x_1,x_2) = \psi_1(x_1)\psi_2(x_2) + \psi_2(x_1)\psi_1(x_2)$$ . Why is it so? Why is it the sum of the two states? What is the ...
6
votes
2answers
417 views

Why do we must initially assume that the wavefunction is complex?

The sound waves are real, and they can interfere, so corresponding apparat may be used in quantum mechanics. We also may use the time dependence in a form of orthogonal matrix multiplying the initial ...
1
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1answer
85 views

Analytical non-separable solution for schrodinger equation

I am an undergraduate with the background of a first course in Quantum Mechanics. I want to find out if there exist non-unique solutions to Schrodinger equation. So I have to find potentials (...
2
votes
1answer
362 views

What is the difference between real orbital & complex orbital?

While reading Atomic orbitals, I came before these two terms. The 'real orbital' is given here: Real orbitals An atom that is embedded in a crystalline solid feels multiple preferred axes, ...
0
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2answers
375 views

Indistinguishable particles and probability density

I am given the following (probably simple) exercise, but I think I misunderstand something: Let $\psi_{a,b}(r_1,r_2)$ be a two-particle state, calculate the probability density for distinguishable ...
-4
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1answer
92 views

Where do these two equalities for the expectation value come from precisely? Doesn't $\Psi^* x \Psi = x |\Psi|^2$?

Where do these two equalities for the expectation value come from precisely? : $$\begin{align} \langle x\rangle &= \int_{-\infty} ^\infty \Psi^* x \Psi\,\mathrm{d}x \\ \langle x^2\rangle &= \...
0
votes
1answer
70 views

The Tunnelling Man [closed]

What's the probability that I will tunnel through a solid wall? By "tunnel" I mean, the probability of finding me on the other side of the wall. Assumptions Wall thickness = $d$ Clearly state any ...
1
vote
1answer
87 views

Mach-Zehnder interferometry wave functions

Consider the set up below: I have read that in the apparatus the wavefunction is given by: $$|\psi \rangle=e^{i\theta}|c \rangle +i |b \rangle$$ where $\theta$ is the phase added by the phase adder....
0
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2answers
100 views

Importance of bound states

While solving a potential well problem we get scattering states and bound states (if exist). Number of the bound states we get depends on the potential profile. What I want to ask is, what is the ...