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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Physical meaning of quantum operators

Let's say we have a wavefunction $\psi$ and a measurement operator $\hat A$. I understand how eigenvalues and eigenvectors of $\hat A$ describe the possible outcomes of the measurement. I also ...
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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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Non-separable solution for the Schrödinger equation

Schrödinger solutions are usually if not always of the type: $\psi=\operatorname{T}(t)*\operatorname{X}(x)$ (we use the separation of variables method to arrive at the time independent Schroedinger ...
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Status of the wavefunction in QM after QFT?

In QFT which is the continuation of QM the Quantum fields are considered as the final objective reality. If this is true why is there not a last and deceisive Interpretation of the wavefunction, which ...
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How do we choose the standard probability current?

In quantum mechanics, the probability current is defined as $$\mathbf{J} \propto \text{Im}(\psi^* \nabla \psi)$$ and satisfies the continuity equation $$\nabla \cdot \mathbf{J} = - \frac{\partial \...
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The explicit solution of the time-dependent Schrödinger equation for a free particle that starts as a delta function

A previous thread discusses the solution of the time-dependent Schrödinger equation for a massive particle in one dimension that starts off in the state $\Psi(x,0) = \delta(x)$. This can easily be ...
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What is the difference between a quantum mechanical wave and a classical wave?

As we know we all say that quantum mechanics is "wave mechanics", and particles are described as waves or associated with every particle a wave nature; the behavior of such waves are described by ...
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How to guarantee square integrable solutions to time-independent Schrödinger's equation?

Given the time-independent Schrödinger’s equation in one dimension $$H\psi = E\psi$$ what restrictions can we place on V(x) (inside the hamiltonian) and E to guarantee that the solutions won't have ...
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Relation between p+ip wave Superconductor and Moore-Read State

I am quite interested in the understanding of the relation between p_ip wave superconductor(SC) and the Moore-Read(MR) state. They share many similar properties, for example, p+ip SC has majorana as ...
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Which position and momentum distributions arise from some wave function?

Consider a particle in one dimension with wave function $\psi$. The probability density function describing how likely it is to find it in a given position is given by $f(x)=\left|\psi(x)\right|^2$. ...
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How do we know that the two indistinguishable particles in the same infinite well have different energies?

I'm reading an example in which we have two identical particles in the same infinite well. They have different quantum numbers "n", which means that they have different energies. This example is used ...
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Hydrogen radial wave function infinity at $r=0$

When trying to solve the Schrödinger equation for hydrogen, one usually splits up the wave function into two parts: $$\psi(r,\phi,\theta)= R(r)Y_{\ell,m}(\phi,\theta).$$ I understand that the radial ...
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Is continuity of the wavefunction “put in by hand” for the Dirac delta potentials?

In 1d, for $V(x) = g\delta(x)$, integrating the TISE yields (assuming that $\psi$ is bounded$^\dagger$, so as to suppress the term containing $E$) $$ -\frac{\hbar^2}{2m} \left( \psi'(\varepsilon) - \...
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How does a Wavefunction collapse?

How does a wavefunction collapse into one state? More specifically, what conditions cause a wavefunction for a quantum particle to collapse? Does this have to do with density matrices? Please ...
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Am I missing a trick to solving a 3D potential well problem?

I was playing around with a 3-D potential $V$ such that $V_{(r)} = 0$ for $r<a$, and $V_{(r)} = V_0>0$ otherwise. By using the Schrödinger Equation, I showed that: $$\frac{-\hbar}{2m}\frac{1}{r^...
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Why is the word 'simultaneously' important in stating Heisenberg's uncertainty principle?

The Heisenberg's uncertainty principle states that a particle cannot have a precise value of its position and conjugate momentum simultaneously. If these uncertainties are intrinsic properties of a ...
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Does the Hilbert space include states that are not solutions of the Hamiltonian?

I've studied Quantum Mechanics and I know the usual answer "The dimension of the Hilbert Space is the maximum number of linear independent states the system can be found in". There is something about ...
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Example of the time-independent Schrödinger equation having a complex solution?

We know $\Psi(x,t)$ is complex, but can $\Psi(x)$ be complex? I have seen particle in a box, well and harmonic oscillator. All have real solutions for time-independent Schrödinger equation. Hence, I ...
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Bra ket notation rigorous way

I'm struggling to see how $\langle x|\Psi\rangle =\Psi(x)$. I have read a few previous bra ket questions in here but still not clear. Any good book for understand the bra-ket notation in more rigorous ...
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Infinite Wells and Delta Functions

In considering a delta potential barrier in an infinite well, I can just enforce continuity at the potential barrier-it doesn't have to go to zero. Why then does it need to go to zero at the walls of ...
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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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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 ^...
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Is the wave function objective or subjective?

Here is a question I am curious about. Is the wave function objective or subjective, or is such a question meaningless? Conventionally, subjectivity is as follows: if a quantity is subjective then ...
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How does gravity affect the wavefunction of a particle?

I'm wondering how gravity affects the wave function of a particle. For example, if we shot a particle horizontal to the earth at a vertical detector screen, would the distribution on the screen be ...
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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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Orthonormality of Radial Wave Function

Is the radial component $R_{n\ell}$ of the hydrogen wavefunction orthonormal? Doing out one of the integrals, I find that $$\int_0^{\infty} R_{10}R_{21}~r^2dr ~\neq~0$$ However, the link below says ...
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Hong-Ou-Mandel Effect for incoherent photons

I am trying to show that the Hong-Ou-Mandel effect will not happen if the photons are not coherent. I am not sure where I am going wrong with my maths. I have tried to emulate the coherent case but ...
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What is the physical significance of the Dual Space in QM?

In studying the math of Quantum Mechanics, I came across the idea of the Dual Space, and how we need it to take scalar products. If a Hilbert space of kets represents state vectors/wavefunctions, what ...
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Can I differentiate a wavefunction with respect to a quantum number?

So I was reading some papers, mainly in the Green's functions theory of the time-independent Schroedinger equation, and came across an equation that had a term similar to: $$\frac{\partial \Psi_n^*(x)...
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Why does the wavefunction have to be continuous in the presence of a Dirac delta potential?

Considering the time-independent Schrödinger equation, I can see for a finite potential, why the wavefunction has to be continuous, I can also see why the first derivative of the wavefunction is ...
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Virial theorem and variational method: a question

I have an hydrogenic atom, knowing that its ground-state wavefunction has the standard form $$ \psi = A e^{-\beta r} $$ with $A = \frac{\beta^3}{\pi}$, I have to find the best value for $\beta$ (...
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Can a measurement partially “collapse” a wavefunction?

Let's say I have a wavefunction $\Psi$ which can be decomposed into a sum of it's energy eigenstates: $$ \Psi = a|1\rangle + b|3\rangle + c|8\rangle + d|10\rangle$$ Where, of course, $|a|^2 + |b|^2 +...
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Physical position eigenfunction normalisation

We know that the Dirac function $$\delta(a)=\lim_{a \rightarrow 0} \delta_{a}(x)$$ can be written as an infinitesimally narrow Gaussian: $$ \delta_{a}(x) := \frac{1}{\sqrt{2\pi a^2}}e^{-x^2/2a^2}$$ ...
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How to interpret a wavepacket in quantum field theory: is it one particle or a superposition of many?

In 'classical' quantum mechanics, a wave packet is a (more or less) localized particle. The wave packet can be expanded in a superposition of plane waves, each with a defined momentum and energy. This ...
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Justification of the energy and momentum operators in quantum mechanics [duplicate]

There is one thing that always troubled me in quantum mechanics, how do you justify the expression of the energy and momentum operators, namely $\hat{E} = i\hbar\frac{\partial}{\partial t}$ and $\...
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Is there a direct physical interpretation for the complex wavefunction?

The Schrodinger equation in non-relativistic quantum mechanics yields the time-evolution of the so-called wavefunction corresponding to the system concerned under the action of the associated ...
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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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Why does a delta-function well have only 1 bound state?

From Griffiths, Introduction to Quantum Mechanics, pg. 73: Evidently, the delta-function well, regardless of its "strength" $\alpha$, has exactly one bound state $$\psi(x) = \frac{\sqrt{m \...
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Something special about energy eigenstates when it comes to time evolution?

A particle is subject to an infinite square well potential with $$V(x)= \begin{cases} 0 & −a \lt x \lt a\\ \infty & \,\,\,\,\text{otherwise} \end{cases}$$ At a time $t=0$ its ...
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What is the relation between position and momentum wavefunctions in quantum physics?

I have read in a couple of places that $\psi(p)$ and $\psi(q)$ are Fourier transforms of one another (e.g. Penrose). But isn't a Fourier transform simply a decomposition of a function into a sum or ...
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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 $y(...
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In Bohmian mechanics, do electrons move inside an atom?

Look at http://www.bohmian-mechanics.net/whatisbm_pictures_hydrogen.html. It is mentioned that in the rest states of a bound electron, the position of the electron is stationary, since the ...
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Why are wave functions required to vanish at infinity?

I'm taking an introductory quantum mechanics class and although we require the wave function to (rapidly?) decay at infinity, I'm not entirely sure why. I have no background in physics (I took AP ...
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Imaginary Eigenvalue Of A Hermitian Operator

The eigenfunctions of a Hermitian operator are real. But consider a function $\psi(x)=e^{-\kappa x}$, $x\in\mathbb{R}$, where $\kappa$ is a real constant. Then, $$\hat p \psi(x)=-i\hbar \frac{d}{dx}e^{...
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Inexact measurement and wavefunction collapse

As is usually said, measurement of an observable $q$ leads to collapse of wavefunction to an eigenstate of the corresponding operator $\hat q$. That is, now the wavefunction in $q$ representation is $\...
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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 $n$ particles,...
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Uniqueness of quantum ladder for the harmonic oscillator

Context: Griffith's book on Quantum Mechanics (QM), in Section 2.3.1, tries to solve for the stationary states $\psi(x)$ of a harmonic oscillator by solving the Time-Independent Schrodinger Equation (...
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Atomic orbitals and complex wavefunction

I have read different questions related to the atomic orbitals labelled with 2px and 2py present here, such as What is the difference between real orbital & complex orbital? or Notation of complex ...
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Why doesn't there exist a wave function for a photon whereas it exists for an electron?

A photon is an excitation or a particle created in the electromagnetic field whereas an electron is an excitation or a particle created in the "electron" field, according to second-quantization. ...
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Sinusoidal to complex form of wave equation

I know that a sinusoidal plane wave can be represented by the wave equation $$ \psi (x,t)=A\, \cos(kx-\omega t) $$ I have also seen that a plane wave can be represented in complex exponential form as $...

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