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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Nodes and Antinodes for standing wave

In the arrangement shown in the figure below, an object of mass m can be hung from a string (linear mass density $\mu$ = 2.00 g/m) that passes over a light (massless) pulley. The string is connected ...
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Vector representation of wavefunction in quantum mechanics?

I am new to quantum mechanics, and I just studied some parts of "wave mechanics" version of quantum mechanics. But I heard that wavefunction can be represented as vector in Hilbert space. In my eye, ...
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464 views

Why does a plane wave have definite momentum?

Apologies if this is a little vague. It might not have a good answer. Given the interpretation of $|\psi(x)|^2$ as a probability distribution it's unsurprising that a wave function that is ...
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349 views

Potential step and its transmission / reflection

Lets say we have a potential step with regions 1 with zero potential $W_p\!=\!0$ (this is a free particle) and region 2 with potential $W_p$. Wave functions in this case are: \begin{align} ...
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184 views

How does one find the wave velocity and the phase speed?

While I was studying beats, I tried to find a displacement function of any particle in the most generalized form. I ended up with $$y=2A\sin(\pi(t-x/v)(f_1+f_2))\cos(\pi(t-x/v)(f_1-f_2)).$$ Now, ...
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Why the hydrogen radial wave function is real?

Why the hydrogen radial wave function is real? Is it a coincidence?
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Why do people say the phase oscillates in time and the amplitude stays the same but the intensity of a traveling beam does oscillate with time?

I'm confused why people say the phase oscillates in time and the amplitude stays the same (the reason for having complex numbers). But on the other hand, the intensity of a traveling beam does ...
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145 views

Why does the wave description say that probability oscillates, while the phase interpretation says constant amplitude?

The wave description of a particle illustrates an oscillating probability of the particle being found in any point in space. When a particle travels, it carries along with it a phase that oscillates ...
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155 views

normalizing a wavefunction

I have a homework problem that I can't get started on, below is the first bit. I feel like I should just be able to integrate to find $C$ but I get a divergent integral. Can someone give me a hint as ...
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Is there anything to prevent paired-up neutrons from a complete overlap

The reason "neutrons don't overlap", as DarenW explained it, has to do with intricate forces at play that take into account the spins, iso-spins and symmetry of the wavefunctions. However, assume I ...
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Why is the Horizontal Force Constant in Deriving the One Dimensional Wave Equation

My textbook in deriving the wave equation for a one dimensional elastic string stated that the horizontal direction force is constant.I understand that the horizontal components of the tensions on ...
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598 views

Finite, square, potential well

Lets say we have a finite square well symetric around $y$ axis (picture below). I know how and why general solutions to the second order ODE (stationary Schrödinger equation) are as follows for ...
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205 views

Is normalization consistent with Schrodinger's Equation?

Schrodinger's Equation does not set a limit on the size of wave functions but to normalize a wave function a limit must be set. How is this consistent physically and mathematically with Schrodinger's ...
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120 views

Phase shift in electromagnetic potential

In Aharonov-Bohm effect, how to derive that the wave function of a electric charge $q$ acquires a phase shift $\phi=\frac{q}{\hbar}\int \mathbf{A} \cdot d\mathbf{x}$ after travelling in the non-zero ...
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89 views

Question about the linearity of wave functions

For piece-wise constant potential, the potential energy is constant so the time dependent wave function can take the form $\psi(x,t)=C_1e^{i(kx- \omega t)}+C_2e^{i(-kx-\omega t)}$ where ...
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130 views

Where is a particle bound in a delta potential?

I can picture a bound state in a harmonic oscillator, or in an infinite square well, but where is a particle bound in a delta potential?
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312 views

Wavefunction restrictions of odd potentials

So I was just reading back through Griffiths' "Introduction to Quantum Mechanics" and solving some of the problems for practice. There is a nice one (problem 2.1c for those playing at home) where you ...
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224 views

What does it mean for something to be a ket?

Ok so I will provide the following example, which I am choosing at random from Sabio et al(2010): $$\psi(r,\phi)~=~\left[ \begin{array}{c} A_1r\sin(\theta-\phi)\\ ...
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Wave function and Dirac bra-ket notation

Would anyone be able to explain the difference, technically, between wave function notation for quantum systems e.g. $\psi=\psi(x)$ and Dirac bra-ket vector notation? How do you get from one to the ...
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296 views

Cylindrical wave

I know that a wave dependent of the radius (cylindrical symmetry), has a good a approximations as $$u(r,t)=\frac{a}{\sqrt{r}}[f(x-vt)+f(x+vt)]$$ when $r$ is big. I would like to know how to deduce ...
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340 views

What does the wavefunction of atom look like at low temperature?

I am reading an introduction material on Bose-Einstein condensation (BEC) at low temperature and it stated that when the temperature approaches zero kelvin, almost all atoms are degenerated into the ...
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Gaussian wave packet

At our QM intro our professor said that we derive uncertainty principle using the integral of plane waves $\psi = \psi_0(k) e^{i(kx - \omega t)}$ over wave numbers $k$. We do it at $t=0$ hence $\psi = ...
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186 views

Once I have the eigenvalues and the eigenvectors, how do I find the eigenfunctions?

I am using Mathematica to construct a matrix for the Hamiltonian of some system. I have built this matrix already, and I have found the eigenvalues and the eigenvectors, I am uncertain if what I did ...
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In Dirac notation, what do the subscripts represent? (Solution for particle in a box in mind)

So the set of solutions for the particle in a box is given by $$\psi_n(x) = \sqrt{\frac{2}{L}}\sin(\frac{n\pi x}{L}).$$ In Dirac notation $<\psi_i|\psi_j>=\delta_{ij}$ assuming $|\psi_i>$ ...
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In the expansion of the scattered wave function, why do these two functions have the same index?

See Griffiths Quantum Mechanics, eq. 11.21. Evidently, $$\psi(r,\theta,\phi)=Ae^{ikz}+A\sum\limits_{l,m}^{\infty}C_{l,m}h_{l}(kr)Y_{l}^{m}(\theta,\phi).$$ But I don't see why the $l$th Hankel function ...
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Could quantum mechanics work without the Born rule?

Slightly inspired by this question about the historical origins of the Born rule, I wondered whether quantum mechanics could still work without the Born rule. I realize it's one of the most ...
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Normalisation factor $\psi_0$ for wave function $\psi = \psi_0 \sin(kx-\omega t)$

I know that if I integrate probabilitlity $|\psi|^2$ over a whole volume $V$ I am supposed to get 1. This equation describes this. $$\int \limits^{}_{V} \left|\psi \right|^2 \, \textrm{d} V = 1\\$$ ...
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Young experiment: square of classical real wave function

I can't understand why the sum of two real waves result in a time dependent wave, but not so for the complex waves. In details, I can't get this passage on p.38-39 in A.C. Phillips, Introduction to ...
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Confused over complex representation of the wave

My quantum mechanics textbook says that the following is a representation of a wave traveling in the +$x$ direction:$$\Psi(x,t)=Ae^{i\left(kx-\omega t\right)}\tag1$$ I'm having trouble visualizing ...
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Amplitude of Probability amplitude. Which one is it?

QM begins with a Born's rule which states that probability $P$ is equal to a modulus square of probability amplitude $\psi$: $$P = \left|\psi\right|^2.$$ If I write down a wave function like this ...
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Probability and probability amplitude [duplicate]

What made scientists believe that we should calculate probability $P$ as the $P = \left|\psi\right|^2$ in quantum mechanics? Was it the double slit experiment? How? Is there anywhere in the ...
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Solving the time independent Schrodinger equation: Does a complex solution make sense?

In my notes, I have the Time Independent Schrodinger equation for a free particle $$\frac{\partial^2 \psi}{\partial x^2}+\frac{p^2}{\hbar^2}\psi=0\tag1$$ The solution to this is given, in my notes, ...
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Superconducting Wavefunction Phase (Feynman Lectures)

In Volume 3, Section 21-5 of the Feynman lectures (superconductivity), Feynman makes a step that I can't quite follow. To start, he writes the wavefunction of the ground state in the following form ...
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Expectation values-Wavefunction

I'm a bit puzzled about an excercise in which I have to find the expectation values for position and momentum. Normally this should be pretty easy but in this case I just don't get the point. ...
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Meaning of $\int \phi^\dagger \hat A \psi \:\mathrm dx$

While analysing a problem in quantum Mechanics, I realized that I don't fully understand the physical meanings of certain integrals. I have been interpreting: $\int \phi^\dagger \hat A \psi ...
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359 views

Is this interpretation of $\psi=\frac{1}{\sqrt{\pi a^{3}}}e^{-r/a}$ correct?

Apologies if this is stating the obvious, but I'm a non-physicist trying to understand Griffiths' discussion of the hydrogen atom in chapter 4 of Introduction to Quantum Mechanics. The wave equation ...
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What does the quantum state of a system tell us about itself?

In quantum mechanics, quantum state refers to the state of a quantum system. A quantum state is given as a vector in a vector space, called the state vector. The state vector theoretically ...
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Does quantum mechanics allow faster than light (FTL) travel?

Let's suppose I initially have a particle with a nice and narrow wave function[1] (I will leave these unnormed): $$e^{-\frac{x^2}{a}}$$ where $a$ is some small number (to make it narrow). Let's also ...
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70 views

What methods exist for us to measure the position and momentum of atoms that make up molecules?

In this paper, Localization of an atom by homodyne measurement. A. M. Herkommer et al. Quantum Semiclass. Opt. 8 no. 1, p. 189 (1996) (paywalled). the authors are able to localize atoms using ...
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Was uncertainty principle inferred by Fourier analysis?

I would like to know: did Heisenberg chance upon his Uncertainty Principle by performing Fourier analysis of wavepackets, after assuming that electrons can be treated as wavepackets?
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Interpretation of $e|\psi|^2$ as electron density

In solid state physics the electron density is often equated to $e|\psi|^2$. However, the Sakurai says (Chapter 2.4, Interpretation of the Wave Function, p. 101) that adopting such a view leads "to ...
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What's the physical significance of the inner product of two wave functions in quantum region?

I am a reading a book for beginners of the quantum mechanics. In one section, the author shows the inner product of two wave functions $\langle\alpha\vert\beta\rangle$. I am wondering what's the ...
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Non-Degeneracy of Eigenvalues of Number Operator for Simple Harmonic Oscillator [duplicate]

Possible Duplicate: Proof that the One-Dimensional Simple Harmonic Oscillator is Non-Degenerate? I'm trying to convince myself that the eigenvalues $n$ of the number operator ...
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159 views

How is wavefunction probability redistributed after partial wavefunction collapse?

Suppose I set up the double-slit experiment using photons as my particle. Behind the left slit I place a beam splitter that points some of the light off in the direction of a camera (represented as ...
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Wavefunction collapse and gravity

If gravity can be thought of as both a wave (the gravitational wave, as predicted to exist by Albert Einstein and certain calculations) and a particle (the graviton), would it make sense to apply ...
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Confused over the presence of 2 expressions for $\Psi(x,t)$

I'm following Griffiths' Introduction to Quantum Mechanics, and I see that he's got 2 different expressions for $\Psi(x,t)$. One of them is ...
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Plane wave expansion in cylindrical coordinates

I am trying to solve scattering problem in 2D and got to expand the wave function in cylindrical system which comes out to be Hankel function. Can you tell me how to expand the plane wave $\exp(i ...
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Does quantum mechanics predict instantaneous action at a distance even without entanglement?

The suggestion that quantum mechanics implies that instantaneous action at a distance occurs is normally based on the contention that this follows from the entanglement of particles that share a ...
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Quantum Mechanics - The Normalization of $\psi_{3,1,1}$

Show that the hydrogen atomic wavefunction $\psi_{3,1,1}$ is normalized, and that it is orthogonal to $\psi_{3,1,−1}$. I'm not sure if I'm supposed to consider the radial part. I can show that ...
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Meaning of instantaneous probability densities in time dependent wavefunctions

For a time dependent wavefunction, are the instantaneous probability densities meaningful? (The question applies for instances or more generally short lengths of time that are not multiples of the ...