Questions tagged [wavefunction]

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. DO NOT USE THIS TAG for classical waves.

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About the complex nature of the wave function?

1. Why is the wave function complex? I've collected some layman explanations but they are incomplete and unsatisfactory. However in the book by Merzbacher in the initial few pages he provides an ...
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Does Heisenberg's uncertainty under time evolution always grow?

Recently there have been some interesting questions on standard QM and especially on uncertainty principle and I enjoyed reviewing these basic concepts. And I came to realize I have an interesting ...
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Normalizable wavefunction that does not vanish at infinity

I was recently reading Griffiths' Introduction to Quantum Mechanics, and I stuck upon a following sentence: but $\Psi$ must go to zero as $x$ goes to $\pm\infty$ - otherwise the wave function would ...
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Why doesn't the nucleus have “nucleus-probability cloud”?

While deriving the wave function why don't we take into the account of the probability density of the nucleus? My intuition says that the nucleus is also composed of subatomic particles so it will ...
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Hilbert space vs. Projective Hilbert space

Hilbert space and rays: In a very general sense, we say that quantum states of a quantum mechanical system correspond to rays in the Hilbert space $\mathcal{H}$, such that for any $c∈ℂ$ the state $\...
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What is the probability for an electron of an atom on Earth to lie outside the galaxy?

In this youtube video it is claimed that electrons orbit their atom's nucleus not in well-known fixed orbits, but within "clouds of probability", i.e., spaces around the nucleus where they can lie ...
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What's the point of Pauli's Exclusion Principle if time and space are continuous?

What does the Pauli Exclusion Principle mean if time and space are continuous? Assuming time and space are continuous, identical quantum states seem impossible even without the principle. I guess ...
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What is a wave function in simple language?

In my textbook it is given that 'The wave function describes the position and state of the electron and its square gives the probability density of electrons.' Can someone give me a very simple ...
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What exactly is deterministic in Schrödinger's equation?

I have read the following on Wikipedia but I can't understand it: In quantum mechanics, the Schrödinger equation, which describes the continuous time evolution of a system's wave function, is ...
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The formal solution of the time-dependent Schrödinger equation

Consider the time-dependent Schrödinger equation (or some equation in Schrödinger form) written down as $$ \tag 1 i \partial_{0} \Psi ~=~ \hat{ H}~ \Psi . $$ Usually, one likes to write that it has a ...
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What is the Copenhagen interpretation of quantum field theory?

I am most interested in interpretational differences due to the fact that quantum field theory is relativistic while quantum mechanics is not. By "Copenhagen interpretation" I mean a minimal ...
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Can someone provide a physical — not mathematical — intuition for the phase in a quantum wavefunction?

I've read every thread on StackExchange (and Quora and reddit...) that I can find about a physical intuition for the phase in the quantum wave function, and I still Just. Don't. Get. It. (Yes, I've ...
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The importance of the phase in quantum mechanics

In introductory quantum mechanics I have always heard the mantra The phase of a wave function doesn't have physical meaning. So the states $| \psi \rangle$ and $\lambda|\psi \rangle$ with $|\lambda|...
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Is the probability of an electron being somewhere zero?

So recently I've been reading "How to teach Quantum Mechanics to your Dog" by Chad Orzel. In chapter 3, he says, if I understood this right, that electrons can only exist in specific quanta - that is ...
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Why do electrons in an atom occupy only the stationary states?

When we talk about the elementary problems in quantum mechanics like particle in a box, we first calculate the energy eigen-function. Then we say that the most general state is the linear combination ...
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3answers
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The meaning of the phase in the wave function

I have just started studying QM and I got into some trouble understanding something: Let's say there is a wave function of a particle in a 1D box ($0\leq x\leq a$): $$\psi(x,t=0) = \frac{i}{\sqrt{5}}...
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Physical interpretation of complex numbers [duplicate]

Complex numbers are used widely in quantum mechanics and the waveform, is there a physical interpretation of what this means about the structure of the universe? Why is it not used in macro physics? ...
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What exactly is a bound state and why does it have negative energy?

Could you give me an idea of what bound states mean and what is their importance in quantum-mechanics problems with a potential (e.g. a potential described by a delta function)? Why, when a stable ...
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Why the statement “there exist at least one bound state for negative/attractive potential” doesn't hold for 3D case?

Previously I thought this is a universal theorem, for one can prove it in the one dimensional case using variational principal. However, today I'm doing a homework considering a potential like this:$...
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Interpretation of Dirac equation states for moving electron

I try to understand a physical interpretation of the four components of the Dirac 4-spinor for a moving electron (in the simplest case, a plane wave). There is a very good question and answer about ...
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Is it possible to reconstruct the Hamiltonian from knowledge of its ground state wave function?

Is it possible to "construct" the Hamiltonian of a system if its ground state wave function (or functional) is known? I understand one should not expect this to be generically true since the ...
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1answer
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How do I simulate an atom?

Let us assume I wish to simulate a Helium atom, since there does not exist a closed-form solution. However, I presume I would need to simulate the time-dependent Schrodinger wave equation. I would ...
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1answer
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What is known about the hydrogen atom in $d$ spatial dimensions?

In a first (or second) course on quantum mechanics, everyone learns how to solve the time-independent Schrödinger equation for the energy eigenstates of the hydrogen atom: $$ \left(-\frac{\hbar^2}{2\...
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Non-integrable Wavefunctions

Suppose first-order perturbation yields a credible correction to the energy, but a correction to the wave function that's not square-integrable. That can happen, I see no reason why it couldn't. ...
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What is a standing wave?

I'm a highschool sophomore, bear this is mind when answering this question, in other words, the answer doesn't need to be in total layman terms, but it should be understandable by an applied ...
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Meaning of inner product $\langle \vec{r} | \psi(t)\rangle $

I have come across the equation which comes out of the nothing in Zettili's book Quantum mechanics concepts and applications p. 167: $$\psi(\vec{r},t) ~=~ \langle \vec{r} \,|\, \psi(t) \rangle.$$ ...
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“Reality” of EM waves vs. wavefunction of individual photons - why not treat the wave function as equally “Real”?

In thinking how to ask this question (somewhat) succinctly, I keep coming back to a Microwave Oven. A Microwave Oven has a grid of holes over the window specifically designed to be smaller in ...
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1answer
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Interpretation of Dirac equation states

In Pauli theory the components of two-component wavefunction were interpreted as probability amplitudes of finding the particle in particular spin state. This seems easy to understand. But when ...
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Is the wavefunction of particles inside a gas spread or localized?

For an individual free particle that starts localized, the wave function packet spreads over time, so the particle becomes less localized. Suppose now that we have a gas of those particles inside a ...
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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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What's the lowest nuclear charge $Z < 1$ that will support a bound two-electron ion $(Z,2e^-)$?

In my programming project I calculate the minimal energy of an atom with 2 electrons in the $L=0, S=0$ state, using a Hylleraas wave function. The values I find for $Z=2$ (He) and $Z=1$ (H$^-$) are ...
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How does the momentum operator act on state kets?

I have been going through some problems in Sakurai's Modern QM and at one point have to calculate $\langle \alpha|\hat{p}|\alpha\rangle$ where all we know about the state $|\alpha\rangle$ is that $$\...
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What is the Continuity Equation in QM?

I have an exercise for my homework that mentions the "continuity equation". Don't tell me how to solve it please, just tell me what the continuity equation is. I tried googling it but I couldn't find ...
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How much of a wave function must reside inside event horizon for it to be consumed by the black hole?

Related to this question in astronomy SE (https://astronomy.stackexchange.com/q/30611/10813) and in particular my attempt to answer it, I started to wonder which fraction of a waveform (for eg a ...
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Is there a condition of quantum mechanics that forbids Lorentzian distributions?

Imagine a particular potential that allows a superposition of eigenstates such that in space basis the probability density $|\psi(x)|^2$ is a lorentzian (Cauchy) distribution. The properties of the ...
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Galilean invariance of the Schrodinger equation

Is the Schrodinger equation invariant under Galilean transformations? I am only asking this question so that I can write an answer myself with the content found here: http://en.wikipedia.org/wiki/...
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Smoothness constraint of wave function

Is there anything in the physics that enforces the wave function to be $C^2$? Are weak solutions to the Schroedinger equation physical? I am reading the beginning chapters of Griffiths and he doesn't ...
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3answers
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Nonexistence of a Probability for Real Wave Equations

David Bohm in Section (4.5) of his wonderful monograph Quantum Theory gives an argument to show that in order to build a physically meaningful theory of quantum phenomena, the wave function $\psi$ ...
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How can I solve this quantum mechanical “paradox”?

Let a (free) particle move in $[0,a]$ with cyclic boundary condition $\psi(0)=\psi(a)$. The solution of the Schrödinger-equation can be put in the form of a plane wave. In this state the standard ...
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Electrons - What is Waving?

If an electron is a wave, what is waving? So many answers on the internet say "the probability that a particle will be at a particular location"... so... the electron is a physical manifestation of ...
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Is it wrong to talk about wave functions of macroscopic bodies?

Does a real macroscopic body, like table, human or a cup permits description as a wave function? When is it possible and when not? For example in the "Statistical Physics, Part I" by Landau & ...
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Tensor product in quantum mechanics?

I often see many-body systems in QM represented in terms of a tensor products of the individual wave functions. Like, given two wave functions with basis vectors $|A\rangle$ and $|B\rangle$, belonging ...
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1answer
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How fast does a wavefunction vanish at infinity?

When solving one-dimensional quantum mechanical systems, I find myself very confused about the behavior of wavefunctions at infinity. Let's first impose three reasonable constraints: The potential ...
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Non-separable solutions of the Schroedinger equation

I'm studying an undergraduate Quantum Mechanics course and I have some doubts about the solution of the Schroedinger equation by the separation of variables method. If we suppose that the solutions ...
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When does the $n$th bound state of a 1D quantum potential have $n$ maxima/minima?

In Moore's introductory physics textbook Six Ideas that Shaped Physics, he describes a set of qualitative rules that first-year physics students can use to sketch energy eigenfunctions in a 1D quantum-...
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1answer
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Wavefunction “single-valued” vs “up to a phase” ( gauge transformation) Existence of global section in comple line bundle

I always meet the following sayings: in one case people say that wavefunction must be single valued, in another case people say that wavefunction could be up to a phase in the same point if there is a ...
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Wave/particle-duality as result of taking different limits of a QFT

There is an account on dualities in quantum field theories and string theories by Polchinski from last week http://arxiv.org/abs/1412.5704 At the end of page 4, he writes the wave/particle ...
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How does Ehrenfest's theorem apply to the quantum harmonic oscillator?

Ehrenfest's theorem, to my level of understanding, says that expectation values for quantum mechanical observables obey their Newtonian mechanics counterparts, which means that we can use newton's ...
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Does the time-independent Schrodinger equation in 1D have an exact and general solution?

The (time-independent) Schrödinger equation is for sure the most important equation in quantum mechanics: $$-\frac{\hbar^2}{2m}\nabla^{2}\psi(\vec{r}\,)+V(\vec{r}\,)\psi(\vec{r}\,)=E\,\psi(\vec{r}\,).$...
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If a wave function collapses into one state, does it ever go back to a superposition of states?

It is my understanding that after an observation, the wavefunction collapses to one state. Thus, if you do an observation right after an observation (that collapsed the wavefunction), you get the same ...

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