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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3D Quantum harmonic oscillator

For an isotropic 3D QHO in a potential $$V(x,y,z)={1\over 2}m\omega^2(x^2+y^2+z^2).$$ I can see by independence of the potential in the $x,y,z$ coordinates that the solution to the Schrodinger ...
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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 ...
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Linear vs. quadratic dispersion relation

In wave mechanics the dispersion relation between frequency $\omega$ and wave number $k$ is linear: $$\omega_n=c k_n$$ But in quantum mechanics, based on Schrödinger's equation, one can show that we ...
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Why can we leave off half of the general solution?

In these pdf notes, it says at the bottom of the first page and beginning of the second: [...] whose solution is: $$\Psi(\theta) = c_1 e^{i\omega\theta} + c_2 e^{-i\omega\theta}$$ Since we are ...
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What is the physical reason behind linearity of Schrodinger's equation?

What is the physical reason for Schrodinger equation to be linear? Though in physics many interactions or dynamics are found non linear.
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Can a wave possess spin?

Since a matter wave is associated with a particle in quantum mechanics, does the wave spins? I mean, can we visualize the spinning of wave or is it possible that the wave spins?
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Comparison of 1D and 3D wave functions

When discussing the Schroedinger equation in spherical coordinates, it is standard practice in QM handbooks to point out that the radial part of the 3-dimensional wave equation bears a strong analogy ...
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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 ...
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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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Born Oppenheimer Approximation: Why can any molecular state be represented as a linear combination of electronic states?

in the Born Oppenheimer Approximation, one expands the molecular wavefunction $\Psi(x,X)$ in terms of the electronic wavefunctions $\phi(x;X)$: $$\Psi(x,X)= \sum_k(c(X)_k\phi(x;X)_k)$$ ($x$ are the ...
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Why is wave function so important?

I am almost sure that the wave function is the most important figures in modern physics book. On the other hand I know that wave function even do not have a physical meaning it self alone! Why is ...
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Where does the wave function of the universe live? Please describe its home

Where does the wave function of the universe live? Please describe its home. I think this is the Hilbert space of the universe. (Greater or lesser, depending on which church you belong to.) Or maybe ...
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Help needed to understand “On the reality of the quantum state”

I am having trouble to understand the reasoning in the following paper, On the reality of the quantum state. MF Pusey, J Barret and T Rudolph. Nature Phys. 8, 475–478 (2012); arXiv:1111.3328. ...
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Why does $\ell=0$ correspond to spherically symmetric solutions for the spherical harmonics?

In quantum mechanics why do states with $\ell=0$ in the Hydrogen atom correspond to spherically symmetric spherical harmonics?
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Historical background of wave function collapse

I wonder what were the main experiments that led people to develop the concept of wave function collapse? (I think I am correct in including the Born Rule within the general umbrella of the collapse ...
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Confusion between the de Broglie wavelength of a particle and wave packets

So I learned that the de Broglie wavelength of a particle, $\lambda = \frac{h}{p}$, where h is Planck's constant and p is the momentum of the particle. I also learned that a quantum mechanics ...
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In interpretations of QM where the wave function is real, what does that mean?

In a lot of interpretations of Quantum Mechanics they believe that the wave function is "real". But what does that mean? Are they saying that the wave function of an elementary particle ...
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Orbital angular momentum of electrons

In a QM class, to study the hydrogen atom, we started by defining the Hamiltonian $H$ for a central potential, then made an orbital angular momentum operator appear as part of $H$, then down the line ...
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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 ...
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Tip of a spreading wave-packet: asymptotics beyond all orders of a saddle point expansion

This is a technical question coming from mapping of an unrelated problem onto dynamics of a non-relativistic massive particle in 1+1 dimensions. This issue is with asymptotics dominated by a term ...
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What does the appearance of a classical particle fundamentally reduce to?

I've been reading an article that describes what seems to be a classical particle as a regularity in the global wavefunction over a quantum configuration space: When you actually see an electron ...
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Rectangular window $\psi$ wave-function and the calculus of $\langle p^2\rangle$ for it

I'm currently considering a rectangular window $\psi$ function: $$ \psi(x) = \begin{cases}\left(2a\right)^{-1/2}&\text{for } |x|<a \\ 0&\text{otherwise.} \end{cases} $$ I am interested in ...
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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 ...
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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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Projection of states after measurement

Continuing from the my previous 2-state system problem, I am told that the observable corresponding to the linear operator $\hat{L}$ is measured and we get the +1 state. Then it asks for the ...
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Probability in Measuring Noncommuting Observables

If I have a particle in a state $\Psi(x) = e^{-x^2}$ could I calculate probability of simultaneously measuring, say, $x > 0, p_x < 0$? I understand that $p_x$ and $x$ don't commute and ...
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Is the expression $S=K \log(\Psi)$ appearing in Schrödinger's first paper well defined?

I am currently reading Schrödinger's papers and happen to have some questions that maybe some expert in the field could clarify for me. Like what happens with $$S = K \log(\Psi)$$ when $\Psi<0$. ...
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Where does either Bohr or Heisenberg mention the idea of the wave function collapsing?

Could someone reference a paragraph written either by Heisenberg or Bohr where they mention the idea of the wave function collapsing?
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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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Young's double slit

Am I right to think the (general) probability distribution of photon in a double slit experiment at the screen has the form $|\psi|^2 = c e^{\alpha x^2}\cos^2(\beta x)$? (Due to the superposition of ...
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Bound states in 1D & 2D [duplicate]

Why does Mother Nature allow bound states in arbitrarily weak attractive potential in 2D but not in 3D? See, for example, this article, arXiv:math-ph/0208011.
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Solution for the Finite 2D Potential Well - Rotational Symmetry [closed]

I was searching for the eigensolutions of the two-dimensional Schrödinger equation $$\mathrm{i}\hbar \partial_t \mid \psi \rangle = \frac{\mathbf{p}^2}{2m_e}\mid \psi \rangle + V \mid \psi ...
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Is there any operator behind probability, in quantum mechanics?

In Quantum mechanics, the probability of finding a particle at position $x$ is given by $|\psi(x)|^2$, where $\psi$ is the wave function. Wonder what is the operator which gives this probability? Is ...
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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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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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Wavefunction, probability and impossible events

A friend of mine asked me a question, which I considered trivial at first, but after a while gave rise to some doubts. For instance, we have a potential well in 1 dimension defined by $$ V(x)= ...
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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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Can a wavefunction be solved to any arbitrary precision, given enough computer time?

I learned that the wavefunction for the hydrogen atom can be solved analytically (we did the derivation in class), but that for more complicated atoms it is "impossible" to solve and that only ...
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Simple Quantum Mechanics question about the Free particle, (part1)

I am reading Introduction to Quantum Mechanics by David Griffiths and I am in Ch2 page 59. He starts out writing the time dependent Schrödinger equation and the solution for $\psi(x,t)$ for the free ...
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Sinusoidal Wave Displacement Function

I am learning about waves (intro course) and as I was studying Wave Functions, I got a little confused. The book claims that the wave function of a sinusoidal wave moving in the $+x$ direction is ...
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Wavefunctions in different Hilbert spaces

The state of a quantum system is represented by a wavefunction usually in some specific Hilbert space, .e.g of position, spin, momentum etc. But before deciding in which of these bases to decompose ...
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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 ...
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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 $e^{-iEt/h}\psi(x)$.
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Why does the magnitude squared of the wave function give us the probability density? [duplicate]

My question doesn't go much beyond the title: Why does $$\left | \psi \left ( x,t \right ) \right |^{2}$$ give us the probability density of something appearing at a certain location? I understand ...
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A misunderstanding regarding infinite square well

Here is a picture of the energy states of infinite potential well. We can see That the first level have a half wavelength which fittes with a full wave of the second level. $$\frac{ \lambda _{1} ...
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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 $\Psi^*$ mean in Schrodinger's formulation of Quantum Mechanics?

I am not a physics student. In one of my courses, some fundamental concepts of Quantum mechanics were needed, so I was going through them when I stumbled upon this. It says $$\text{probability} = ...
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Does $\lvert\langle p\lvert\psi\rangle\rvert^2$ have any meaning at all?

I used to think $\lvert\langle p\lvert\psi\rangle\rvert^2$ had the meaning of some likelihood of the particle's momentum being $p$ (within some tolerance interval $\Delta p$). Now I'm just confused. ...
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What is the analogy of $|x\rangle$ in quantum field theory?

Let me start from path integral formulation in quantum mechanics and quantum field theory. In QM, we have $$ U(x_b,x_a;T) = \langle x_b | U(T) |x_a \rangle= \int \mathcal{D}q e^{iS} \tag{1} $$ ...
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Can the expectation value of the square of momentum be negative?

I've been solving a problem in quantum mechanics, and I was deriving the standard deviation of $P$, knowing that $\langle P\rangle=0$. Because $\Delta P=\sqrt{\langle P^2 \rangle - \langle P \rangle ...