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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Calculating the probability of electrophilic attack on terminal C atoms of butadiene using its wavefunction

I'm studying for a test and came across this question: The highest occupied molecular orbital (HOMO) of butadiene is one of the pi orbitals and it has the coefficients: ...
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1answer
100 views

Question about Hartle and Hawking's universal wavefunction?

My apologies in advance if this question is poorly worded or doesn't make any sense, however I have just finished reading into this theory and it seems as though Hawkings No Boundary Universe is ...
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31 views

How to calculate the expectation value of position vector?

$$\psi (\vec{x})=Ae^{-(1/4a^2)|\vec{x}-\vec{x}_0|^2}e^{i\vec{p}_0\cdot \vec{x}/\hbar}$$ The wave function is like this, then how is the expectation value of position vector (not position) calculated? ...
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161 views

How to find the time evolution for two-component spinor?

Taking the wave function I found here as the initial condition for the Schrodinger equation. I would like to find how the wave function will depend on time, at least for small times, and calculate ...
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31 views

How to solve infinite square well with exponential solution (of oscillatory type)?

Given a potential well of $V = 0$ on the interval $(0,L)$ and $V = \infty$ outside the well, I am working to solve the Time Independent Schrodinger Equation $$\dfrac{d^2}{dx^2} \psi= ...
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Valence bond wavefunction of nitrogen

Could someone explain to me how one finds the valence bond wavefunction of an atom? Take nitrogen for example, I know both nitrogen molecules have a valence-electron configuration of 2s22p1x2p1y2p1z ...
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51 views

Position space wave function of an electron

In Wikipedia I find the wave function of a free particle to be $$ \Psi(\vec{r},t) = A\,e^{i(\vec{k}\vec{r}-\omega t)}$$ This is is a plane wave moving in the direction of $\vec{k}$ with speed (phase ...
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112 views
+50

Eigenvalues of the radial Schrödinger equation on a finite integration interval

There are numerous ways to estimate the eigenvalues of a radial Schrödinger equation, see http://arxiv.org/abs/math-ph/0703040 as an example. Anyhow, the formulas only cover the Schrödinger equations ...
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General question about the potential barrier problem: Why does $\exp( kx)$ diverge when $x>0$ in the case when $E < V(x)$?

For the two images below, the first potential barrier has particles approaching it where $E > V_o$ & the second has a particle that has $E < V_o$, where $E$ is the energy of the particles ...
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52 views

QM: Why is there a minus sign on the Energy operator when using complex conjugate?

I understand how they get the first equation. But I have no idea why there is a minus sign on the second equation: This is from a derivation for the probability density current found here: ...
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171 views

writing wave function of a spinor having a spin component [closed]

The effective Hamiltonian for photonic graphene is given by \begin{equation} H=\hbar\nu_{F} {\bf \sigma}\cdot\left(q+\frac{\Delta \vec S^{\prime}}{\hbar\nu_{F}}\right), \end{equation} where ...
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80 views

Is there something wrong with quantizing two times in second quantization?

Second quantization is sometimes considered to be a bad name, because a single quantization is enough. For electrons, we can either start from a many body viewpoint and introduce field operators or we ...
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53 views

Help in solving Schrödinger equation for Hydrogen

I have almost finished getting the solution to the Schrödinger equation for the hydrogen atom (got the theta and phi component equations), but am stuck on the r component equation. Can anyone help me ...
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1answer
45 views

How do I take take the partial derivatives of the general solution to the TDSE for a free particle? [closed]

Consider the general solution to the time-dependent Schrödinger equation for a free particle \begin{align*} \Psi(x,t) &=\frac{1}{\sqrt{2\pi}}\int_{-\infty}^{+\infty} \phi(k) e^{i\left(\hbar ...
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1answer
34 views

Can someone clarify what should and should not be an operator in my verification of the 1D solution to the SE for a free particle?

I just worked out the 1D free particle solution to the Schrödinger equation. My wave function was \begin{equation} \psi(x,t) = Ae^{i(px-Et)/\hbar} \end{equation} So I plugged this into both sides ...
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2answers
929 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 ...
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1answer
2k views

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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1answer
491 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 ...
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3answers
198 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) ...
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4answers
161 views

Derivation of Schrödinger equation - free particle

I learn quantum physics from Alonso-Finn's book (Amazon link), there's one step of Schrödinger equation for a free particle that I couldn't understand. $$ \frac{\mathrm{d^{2}\Psi } }{\mathrm{d} ...
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1answer
275 views

Integers, Energy levels, and wavenumbers for a particle in a 2D box

(This question is not about coding) I have built a little code in Python that allows the user to plot the energy vs the wave number of particle in a 2D box, depending on what values for the integers ...
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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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3answers
118 views

Justifying the notation $\langle x\ |\ \psi\rangle$ [duplicate]

I came across this expression: $$\langle x\ |\ \psi\rangle=\psi(x)$$ How can it be justified? I understand the LHS as an inner product, and the RHS just as a function of the parameter $x$.
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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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3answers
58 views

Why is the expectation value of ground state electron momentum zero?

I have a normalized energy eigenfunction for the ground state of Hydrogen which is $$ \Psi(r) = \frac{1}{\sqrt{\pi a_0^3}}\exp\left(-\frac{r}{a_o}\right), $$ where $a_o$ is the Bohr radius, I have ...
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129 views

What does “spread of momentum” actually mean?

I was reading Feynman's lecture in which Feynman invoked his own way of explaining the uncertainty principle using single-slit experiment. There I found: To get a rough idea of the spread of ...
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2answers
44 views

Why does a plane wave leave the position of the particle unspecified?

I'm covering a book on QM, and just started recently and I'm stuck at understanding something. It says that we can describe the state of motion of a particle with an infinite plane wave equation: ...
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1answer
139 views

Calculation of the $\langle H \rangle$ for a particle in a box

I am working through a problem in which a particle is in an infinite potential well of length $L$ at $t=0$ before the spontaneous change of the box being expanded to length $2L$. I have calculated the ...
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1answer
133 views

Which position and momentum distributions arise from some wave function?

Consider a particle in one dimension with wave function $\psi(x)$. The probability density function describing how likely it is to find it in a given position is given by ...
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1answer
77 views

Why is the full eigenfunction a product of eigenfunctions and not a sum?

For example suppose there is a two electron system. Why is the full eigenfunction a product of the spatial eigenfunction and spin-wave-function for the two electron system?
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What are the conditions of wave function continuity when solving for Dirac Spinors as done in “Klein paradox” paper by Novoselov?

In the paper "Chiral tunneling and Klein paradox" paper by Katsnelson, Novoselov, and Geim, they use the wave function for Dirac spinors. What are the conditions for continuity of the wave function ...
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109 views

Triangular barrier in infinite potential well

Suppose I am looking to solve the wavefunction for the following 1D potential: $$U(x) = \begin{cases}V_0\frac{a-|x|}{a}&\quad\text{for}\quad|x|<a ...
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1answer
69 views

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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2answers
57 views

The delta function as an eigenfunction of the position operator explanation

$\delta (\textbf{r})$ can be interpreted as a wavefunction. [...] It is non-vanishing only for $\textbf{r}=0$. [...] $\delta(\textbf{r})$ is an eigenfunction of the position operator with ...
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70 views

Normalization of wave function meaning…?

I just have one question. I'm doing a problem where I'm told to normalize a wave function, which is split up into two regions, namely where $r \leq r_0$ and $r > r_0$. My question is, why am I ...
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1answer
34 views

Probability density function of a particle for computation [closed]

I'm writing a program, part of which relies on a particle being able to change location similar to a how a real particle would behave (pardon my physics). For example, on a grid of 100x100, a ...
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1answer
79 views

How does one normalize this wavefunction? [closed]

Here is the question: So I could write $ N = \dfrac{1}{{\sqrt{<Ψ|Ψ>}}} $, right? Considering the parentheses in the exponential term, it looks like a good idea to switch to spherical polar ...
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701 views

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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61 views

Eigenstate vs collapsed wave function

An eigenstate, or determinate state, is a state where the measurement of some observable always yields the same result. This means that the standard deviation of the observable is zero. If a ...
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2answers
107 views

Why do wave functions need to be normalized? Why aren't the normalized to begin with? [duplicate]

Before I started studying quantum mechanics, I thought I knew what normalization was. Just pulling off Google, here's a definition that matches what I've understood normalization to mean: ...
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5answers
290 views

Why do we need a wave function?

Assuming our only aim is to solve double slit experiment (or other problems that can be mapped into that). Knowing that electron does some strange thing not expected of a particle, we need a function ...
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159 views

Problems while numerically computing band structure using k.p theory

I want to use k.p theory to numerically compute the band structure of a bulk semiconductor. The band I like to include are the lowest conduction band (cb), the heavy-hole (hh), the light-hole (lh) and ...
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1answer
49 views

Fermions in a well

I have two identical fermions in an infinite potential well. They are non-interacting. How should I show that the first excited state is four-fold degenerate? Is the wavefunction just the ...
5
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2answers
328 views

Can expectation value be imaginary?

I was solving a problem and the result of the expectation value of an operator came out to be $-\frac{\hbar}{4}$ $i$. Is this result possible? It seems counter intuitive.
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3answers
76 views
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1answer
35 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 ...
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1answer
3k views

How to find the wavefunction that solves an infinite square well with a delta function well in the middle?

Solutions for the wavefunction in an infinite square well with a delta function barrier in the middle are easily found online (see here for an example). I am wondering what the wavefunction is for an ...
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2answers
79 views

What is the necessity of wave packet in studying matter wave?

I am new to this realm of physics. I have literally understood the matter wave, wave function; read the trapped electron in an infinite potential-well. But what I didn't understand is the concept of ...
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Normalizing the solution to free particle Schrödinger equation

I have the one dimensional free particle Schrödinger equation $$i\hbar \frac{\partial}{\partial t} \Psi (x,t) = -\frac{\hbar^2}{2m} \frac{\partial^2}{\partial x^2} \Psi (x,t), \tag{1}$$ with ...
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2answers
95 views

Relation between Wave equation of light and photon wave function?

Suppose in our double slit experimental setup with the usual notations $d,D$, we have a beam of light of known frequency $(\nu)$ and wavelength $(\lambda)$ - so we can describe it as $$ξ_0 = ...