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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15
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4answers
2k views

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 ...
17
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2answers
659 views

The formal solution of the Schrödinger equation

Consider the 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 formal solution ...
0
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1answer
102 views

Wave function of particle and antiparticle

The wave functions of particle and antiparticle are related by complex conjugation and wavefunction $Ψ$ must be complex for particle such as $n$, $p$. Is there way to prove this mathematically? Can we ...
3
votes
2answers
273 views

Logic of the 'imaginary wave function collapse' argument in Double Slit experiment

My question is in regards to the stance that the 'wave function collapse' is not an actual physical occurrence. That is, you are not, by observation, changing the particles position from a wave to a ...
0
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1answer
40 views

Quantization of energy in semi-infinite well

Consider an electron with total energy $E>V_2$ in a potential with $$V(x)= \begin{cases} \infty & x< 0 \\ V_1 & 0< x< L \\ V_2 & x>L \end{cases} $$ ...
2
votes
0answers
106 views

Linear combination of eigenstates problem [closed]

Let's say that we have a system such that $$\Psi(x,0)=\frac{\sqrt3}{2}\phi_1(x)+\frac12\phi_2(x)$$ where both $\phi(x)$ are eigenfunctions of the Hamiltonian operator. I want to find $\Psi (x,t)$ ...
1
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1answer
62 views

Schrödinger's Equation with multi-part potential

I have this potential $$V(x) = \left\{ \begin{array}{ll} \infty & \mbox{if } x < -a \\ \frac{V_o}{a}x & \mbox{if } -a \leq x \leq a \\ V_o & \mbox{if } x \geq a \ \end{...
2
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1answer
139 views

Modern interpretation of wave-particle duality

As far as I understand, in the early days of quantum theory there was quite a lot of debate over how to interpret what it meant for a quantum mechanical object to exhibit both wave-like and particle-...
0
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1answer
213 views

Probability and double slit

if a beam of identical particles at random distances from each other (or exactly 1/2 lambda between each other) travelling with the same v towards a double sllit do not interfere with each others wave ...
3
votes
2answers
105 views

Does a wave function describe the motion of electrons or atoms?

I took the course of quantum mechanics a while ago. I do not quite remember all the detail on how to derive the wave function for hydrogen but I still remember the general picture. I think the text ...
-1
votes
1answer
51 views

When does a wavefunction get re-created after a collapse? [closed]

Maybe another way of asking this would be, "When is decoherence un-done for a particle?" Example: Consider that we shoot an electron from a gun. Whilst in transit the electron is just a probability ...
-1
votes
1answer
307 views

Semi-infinite / Asymmetric potential well

I'm asked to come up with an ansatz and solve for the coefficients of a asymmetric infinite potential well, where: $$ V = \begin{cases} \infty \text{ for } x< 0 \\ V_0 \text{ for } 0 \leq x \leq L ...
1
vote
1answer
110 views

Tricky particle in an infinite potential well question

For a particle in an infinite square-well potential in an energy eigenstate, the probability distribution relating to outcomes of position measurements vanishes outside the square well and takes a ...
4
votes
2answers
2k views

Coupled quantum harmonic oscillator

Given the following Hamiltonian for two identical linear oscillators with spring constant $k$ and interaction potential $\alpha x_1x_2$; I was asked to find the expectation value $\langle x_1x_2\...
1
vote
1answer
46 views

Most probable radius of hydrogen in its ground state [closed]

I'm having trouble understanding how we do this. I know we must find the probability density function and then we can optimise it to find the most probable radius. I thought we would just take the ...
1
vote
1answer
47 views

Examples of Matrix Product State(s)

Matrix product states(MPS) is a way of representing a (many-body) wavefunction. The method has been described in, https://arxiv.org/abs/1008.3477 However, would it be possible to see a concrete ...
0
votes
1answer
224 views

How can energy be negative in a finite square well?

Say if the potential $V(x) < 0$ in the well but the sides or the scattered states its zero potential, anyways How is that the energy in the well is less than zero? Is it because the potential ...
3
votes
1answer
647 views

Density of classical states in quantum theory

Let's first treat electrons as classical objects. I can evaluate the classical energy of each state in a configurational space (3N real numbers and, say, spins) using just Coulomb's law. Then I ...
14
votes
7answers
289 views

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 ...
0
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1answer
48 views

Why pressure standing waves have different nodes location with respect to the corresponding displacement waves?

In acoustics the pressure wave has a $\pi/2$ phase difference with the displacement wave. But I do not understand how this leads to a different position of nodes of pressure inside a tube with ...
-1
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1answer
49 views

Infinite square well - periodic boundaries

If we have an infinite square well, I can follow the usual solution in Griffiths but I now want to impose periodic boundary conditions. I have $\psi(x) = A\sin(kx) + B\cos(kx)$ with boundary ...
2
votes
0answers
309 views

Analytic form of the normalization constant for Laughlin wavefunction

Is there any analytic form of the normalization constant for Laughlin wavefunction $$\prod_{i < j} (z_i-z_j)^{1/\nu} e^{-\sum_i |z_i|^2/4}$$ where $\nu$ is the filling factor?
0
votes
1answer
159 views

What's the relation between molecular orbitals and electron density?

The way molecular orbitals are drawn represent the "encapsulated" space in which the wave function has a significant amplitude. How do I obtain from this the electron density? Is there a fundamental ...
0
votes
1answer
25 views

Why must the separation constant be real in a time dependent wave function?

I'm not sure if I'm asking this right. I'm reading ''Introduction to Quantum Mechanics'' by Griffiths and in the chapter 2 exercises he asks to prove that the separation constant, $E$, must be real. ...
0
votes
1answer
75 views

Applying angular momentum operator [closed]

How are the algebraic steps to applied the angular momentum operator defined as: $$\hat{L}=-i\hbar[r\times\nabla]$$ to $$\Psi=a~ \psi_{431}$$ where the $\psi_{nlm}$ are the eigenfunctions of the time ...
-2
votes
0answers
54 views

Expectation value of an Observable and Eigenstates

I am learning about Quantum Mechanics at the moment and I was wondering about Eigenfunctions and Observables. The question I would like to ask is, If a wavefunction is not an eigenstate of an ...
1
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0answers
69 views

How to plot numerically the wave functions according to the Hamiltonian?

It is often difficult to analytically solve the Schrodinger equation, and so we need to obtain a solution numerically. An example plot is shown below. Here, the wave functions for a three junction ...
1
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0answers
54 views

Rotating fermion and spin structure on manifold

We know that doing a 2$\pi$ rotation would give a minus sign to wavefunctions of electrons. Since electrons are spin $1/2$ objects. How is this related to the spin structure on the manifold in which ...
0
votes
0answers
29 views

Conditions to find standing waves harmonics

I came up with a doubt on standing waves conditions. The type of question I find difficult to answer is of the following type. Consider a rope. I do not know if the rope is fixed at both end or at ...
0
votes
1answer
36 views

Situations in which there is path difference interference or formation of standing waves [closed]

I came up with a doubt about standing waves and path difference in general. Consider these two different cases as examples. If I have a rope fixed at one end and I make the free end oscillating, I ...
0
votes
3answers
569 views

Quantum Mechanics in Electric Field

I am working on a problem which looks like this. Consider a charged particle with charge $q$ trapped in a box of length $L$ with finite constant potential $ V_0 $ on both ends. A constant (static) ...
7
votes
4answers
1k views

What does the Schrodinger Equation really mean?

I understand that the Schrodinger equation is actually a principle that cannot be proven. But can someone give a plausible foundation for it and give it some physical meaning/interpretation. I guess I'...
0
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2answers
154 views

General formula for expanding wave function in terms of orthogonal states?

Given a wave function $\psi(x) = \langle \psi | x \rangle$. It can be expanded in terms of orthogonal states: $$ \langle \psi | x \rangle = \sum_n \langle \psi | n \rangle \langle n |x \rangle $$ ...
2
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1answer
74 views

How to understand permutations of particles in Quantum Mechanics?

I'm studying identical particles in Quantum Mechanics and I'm having a hard time to understand the idea of permutations of particles from a mathematical standpoint. From one intuitive point of view ...
0
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1answer
32 views

Normalisation of angular wave function: particle in a circular box

For a particle in a circular box (with radius $R$) with zero potential inside the circle and infinitely high potential outside of the circle, the Schrödinger equation in polar coordinates is: $$-\...
3
votes
1answer
428 views

Double slit experiment and entanglement [duplicate]

Just wondering, what would happen in this experiment. In the experiment you would first have two entangled particles. Then you fire one of the particles, lets say "Particle A", at a double slit ...
0
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0answers
38 views

Applicability of wavefunction matching when Hamiltonian (not just potential) varies

A simple tunnelling calculation can be performed for a potential step by calculating the eigenfunctions for the Hamiltonian on either side of the step and matching the wavefunctions (and using ...
4
votes
2answers
188 views

Gauge transformation of vector potential multiplies wavefunction by phase

Consider an electron in an electromagnetic field with scalar and vector potentials $\phi, \mathbf{A}$. Suppose for simplicity that $\mathbf{A}$ is time independent. Suppose also that we know the ...
0
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0answers
30 views

How can an orbital be recognised from the wavefunction notation?

I am a student and was working up the exercises in my book when I came across a problem that required me to identify the orbital given by $ \psi_{3,2,1}\,.$ What I can work out is that the sub-shell ...
2
votes
2answers
109 views

Why does a electric Potential have to be real, but not a Potential in quantum mechanics?

So I had this Problem when I had to learn about classical electromagnetism: Why is it, that we use complex numbers when calculating stuff, but in the end only the real part is important (for example ...
4
votes
3answers
2k views

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 ...
3
votes
5answers
198 views

What happens when two wavefunctions meet?

Apologies for the over-broad question(s), but I'm having a hard time finding out where to look to answer these myself: If a particle is a wavefunction describing a probability amplitude distributed ...
0
votes
1answer
51 views

Can a quantum mechanical system have more than one wave-function?

I was told that a quantum mechanical system is completely determined by its wave function. But superposition principle says that given two wave functions of some system, a linear combination of them ...
3
votes
1answer
41 views

Protocol for solving time independent Schrodinger equation

Just a short question about the protocol for solving the time-independent Schrodinger equation for different potentials and the reasons for accepting and rejecting solutions. Take for example the ...
5
votes
3answers
137 views

Can a physical wavefunction be non-smooth (its first derivative is discontinuous)?

Here's an argument that might support the statement that such a non-smooth wavefunction is not physical: You cannot add a finite number of smooth functions to get a non-smooth function. By fourier ...
0
votes
0answers
39 views

Energy Conservation in Changing Potential Well

If you prepare a particle in a basis state, $|n\rangle$ of an infinite potential well of length $L$, the energy of that state will be $\langle E\rangle = E_n$, with zero variance. If you then ...
6
votes
2answers
172 views

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: $$ \delta_{a}(x) := \frac{1}{\sqrt{2\pi a^2}}e^{-x^2/2a^2}$$ ...
1
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0answers
51 views

How can we justify identifying the Dirac delta function with the eigenfunction of position? [duplicate]

I can think of at least two different ways to understand eigenfunctions of operators in quantum mechanics. But neither one seems to provide a good explanation for why we take the position-basis ...
1
vote
1answer
61 views

Solution for Schrödinger equation for constant box potential?

It is known that in a box potential, when we set $V = 0$ inside and $V = \infty$ on the boundaries, the solution to the equation $$ - \frac{\hbar}{2m} \bigg( \frac{\partial^2}{\partial x^2} + \frac{ \...
0
votes
2answers
77 views

What do operations on single Qubits of Unfactorable Superpositions Do?

So suppose I have the following Quantum Circuit: A ---- |Control| -----|Hadamard|---- B ---- |xxxxxxx|------------------------ Which is a 2 input Controlled Gate (applying some gate of two choices ...