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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Why must the probability be the integrated square modulus of the wave function [duplicate]

Quantum mechanics uses the wave function to calculate probabilities by taking the square modulus of the wave function as requirement by Max Born. Why should this (squaring of the wave function) be so, ...
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Exact closed form solution to the quantum harmonic oscillator

I came across this question in Griffiths QM, which asked to show that this equation $$\Psi(x,t)=\left(\frac{m\omega}{\pi \hbar}\right)^{1/4} \exp\left[-\frac{m\omega}{2\hbar} ...
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50 views

Scattering off of a bi-local potential

I am trying to figure out the scattering wave function for the following potential: $$V(x,x')=-A \phi(x)\phi^*(x')$$ Such that the SE can be written as $$[\frac{\hbar^2\partial^2_x}{2m}-E]\psi = ...
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Valence bands question

I'm currently doing solid state physics and learning about semiconductors. During the course, I have seen a lot of energy/wavevector graphs, like this one (pic from Kittel): I did not have a ...
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59 views

Does wave function of an electron itself move?

As far as I know quantum mechanics, electrons in an atom in vacuum move accordingly a wave function (a complex scalar field), but the wave function itself does not move (except that the atom may ...
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63 views

Are Neutrons and anti-Neutrons attracted to each other over distance?

Lets create a scenario where you have a total vacuum and you're shooting into this vacuum two streams, one, a Neutron stream and the other an anti-Neutron stream and because you're curious what will ...
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Particle in a box: value for wave function $u(x)$ when potential $V(x)$ is infinity

The time-independent Schrödinger equation (TISE) is: $$ -\frac{\hbar^2}{2m}\frac{d^2 u(x)}{dx^2}+V(x)u(x)=Eu(x) \hspace{15pt}$$ where $E$ is a constant. Imagine now a infinity potential well as ...
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47 views

What is the main difference between a free particle on a line and a free particle on a circle?

The energy spectrum for a free particle in a circle with radius $r$ is $$E_n=\frac{n^2\hbar^2}{2mr^2}.$$ The energy spectrum for a free particle on an infinite line is similar. If so, what is the ...
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60 views

Angular momentum for a given Wavefunction [closed]

Problem: Give a particle in the state $\Psi = e^{\frac{-(x^{2}+y^{2}+z^{2})}{a^{2}}}(\frac{x}{a} + \frac{yz}{a^{2}})$, what are the allowed values for $l_{x}$ (and later for $l_{y}, l_{z}$). Attempt: ...
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110 views

Why are eigenspaces of a Hermitian operator mutually orthogonal? [closed]

In Quantum Mechanics, from the properties of the solution of Schrodinger's Equation inside the infinite well, is that they are: Mutually orthogonal for different eigenvalues. Orthonormal. Complete. ...
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75 views

The boundary condition for delta function

Beginning with the Schr\"odinger equation for $N$ particles in one dimension interacting via a $\delta$-function potential $$(-\sum_{1}^{N}\frac{\partial^2}{\partial ...
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Width of a Gaussian wave function [closed]

Consider the wave function with probably density given in position space as follows: $$P(x,t)=|\Psi(x,t)|^2=\frac{\pi^{-1/2}\Delta p_x/\hbar}{[1+(\Delta ...
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92 views

Particle in a double delta potential, scatter states

I was studying the scatter states of a particle in a double delta potential given in this link: Double delta function well – scattering states But I dont understand how equations (20) and (21) were ...
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64 views

Quantum States, Hilbert Space and Time

I'm having troubles with the assertion "(normalizable) wave-functions constitutes (projective) Hilbert space". The standard argument I find for this seems to go as following: say $\Psi(\vec{x},t)$ is ...
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489 views

Importance of Schrodinger equation [closed]

Louis de Broglie suggested that for microparticles like electrons, wave-like properties can be applied in order to explain some phenomena. Schrodinger wrote down an equation, a wave equation, ...
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70 views

Electrons are 3 dimensional quantized waves (wave functions)?

I thought that electron wave functions were only mathematical of were to find the electron. ...
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53 views

In what cases and with what method does one find a time dependent probability density for a quantum system in an infinite square well?

How can one find the time dependent probability density function of a quantum system given $\Psi(x,t=0)$? Say, $\psi(x) \sim x^4$ for $0 < x < L$. How can one find the time dependant probability ...
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106 views

Relationship between nodes in wavefunction and orthogonality

I read that if I want to construct a wavefunction orthogonal to given $n$ orthogonal wavefunctions, then the new wavefunction should have $n$ nodes. Is this valid under all conditions? Is there a ...
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93 views

Finite square wall with $E > V_0$

I'm working through a problem for homework and feel as if there is a typo or I am confused. The problem is with a one sided finite square wall such as this: So the energy is more than $V_0$. I'm ...
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135 views

What are the functions of these coefficients $c_1,c_2,c_3,c_4$ in $ \psi_{sp^3}= c_1\psi_{2s}+ c_2\psi_{2p_{x}} + c_3\psi_{2p_y}+ c_4\psi_{2p_{z}}$?

Hybridised orbitals are linear combinations of atomic orbitals of same or nearly-same energies. Atomic orbitals interfere constructively or destructively to give rise to a new orbital which is what we ...
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93 views

Is it okay to put singularities into the wave function to test behavior around unstable potentials? [closed]

$$ \psi(r)=\sqrt[4]{\frac{ a}{8\pi^3 }}\frac{ \exp (-a r)}{r^{1.25}} $$ The wave function above is an example of a function that is normalizable in 3D space and $r=\sqrt{x^2+y^2+z^2}$. $$ -\psi ...
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Normalized wave functions in position and momentum space

Using the following expression for the Dirac delta function: $$\delta(k-k')=\frac{1}{2\pi}\int_{-\infty}^{\infty}e^{i(k-k')x}\mathrm{d}x$$ show that if $\Psi(x,t)$ is normalized at time $t=0$, ...
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87 views

A wave function that is normalized initially remains normalized

Suppose that $\Psi(x,t)$ is normalized at time $t=0$. Show that this implies that $\Psi(x,t)$ is normalized at all other times. I know that this makes intuitive sense, and we'd certainly want our ...
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41 views

What is the equation of motion for multiple simultaneous pressure waves in a medium? (In the context of stimulated Brillouin scattering)

My overall motivation is to derive the behavior of Brillouin scattering in a birefringent fiber. Brillouin scattering is a nonlinear interaction between light and sound. In classic Brillouin ...
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87 views

Wave packet and group velocity?

While finding the wave function of a free particle say $\psi(x)$, my textbook says that this is nonphysical because the wave velocity is not same as particle velocity of the free-particle. I don't ...
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Quantum harmonic oscillator - Where am I going wrong?

Find the relationship between $a_+\psi_n$ and $\psi_{n+1}$ My attempt: I was able to prove that $\int{(a_+\psi)^*(a_+\psi)dx} = \int{\psi^*({a_-a_+\psi})dx}\qquad\qquad (1)$ And, ...
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214 views

What is the connection between gravitons and geometry?

I know there are two ways to do quantum gravity. One can pick a background space-time (usually Minkowski flat space-time) and then at any time slice one can define the state of the universe as the ...
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Where do we get an initial wavefunction for a particle or a system?

In many QM text books, a numerical problem starts with the statement that "a particle has an initial wave function of ..." My question is how do we find the initial wave function in the first place?
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35 views

How is charge separation in a polarized vacuum possible?

How is Charge Separation in a Polarized Vacuum possible? The Uehling potential provides the first correction term to the classical coulomb potential from QFT vacuum polarization: $$ V(r) \approx ...
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130 views

Mathematical confusion in quantum mechanics

During a class about Ehrenfest theorem, my teacher use an equation to proceed its derivation (to prove $\frac{d<r>}{dt}=\frac{<p>}{m}$ ) and that is: ...
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81 views

Uncertainity principle and double slit experiment?

My Understanding of uncertainty principle goes that if some particles are in same state, then their measurement of certain property (say $x$ and $p$) will be different for different particles. ...
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98 views

Normalizing a wave function [closed]

A particle with mass $m$ is moving in one dimension. The wave function of the particle is $$\Psi(x,t)=Axe^{-(\sqrt{km}/2\hbar)x^2}e^{-i\sqrt{k/m}(3/2)t}$$ for $-\infty<x<\infty$, ...
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187 views

QM and relative phases

I recently started formally learning about QM. I have studied thus far that any global phase difference is irrelevant when taking energy expectation values. However, that is not evidently the case for ...
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154 views

Where are the worlds in many-worlds interpretation?

What does it mean in MWI for other universes to exist? Are they in some sector of spacetime beyond our cosmic horizon or is it more complicated? I'm not asking this on Philosophy SE because people ...
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148 views

Normalization of potential barrier solution

I don't understand a point in the solution attached to this barrier potential problem. Below equation 4.209, they say Assume first that the wave function on the right side of the barrier in the ...
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1answer
55 views

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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1answer
54 views

Derive probability current density - factors of 2 discrepancy [closed]

To derive the probability current density for a particle in an electromagnetic field, we calculate $\dfrac{\partial \rho}{\partial t} = \dfrac{\partial}{\partial t} (\Psi^* \Psi) = \dfrac{\partial ...
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137 views

Wavefunction interpretations in QM

From two-slit electron-interference experiment we can infer that there is a wave $\psi(x,t)$ that can be associated with electron. The amplitude at some point is the sum of amplitudes reaching that ...
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380 views

What is the wavefunction of the Young Double Slit experiment?

I have never seen the wavefunction for this experiment and would like to know how to derive it using the Schrodinger equation. I specifically want to see how the electron wave function leaves the ...
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70 views

Wave function: what does “1% chance of finding the particle in this area” mean

Say I have 1 electron in some quantum state Defined by some wave function, and it's doing its thing fluctuating the probabilities of where it might be. What if I put a measuring device in an area ...
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285 views

Why are electron wavefunctions standing waves?

How can I convince myself that wavefunctions of electrons on molecular orbitals are indeed standing waves? Is it a consequence of the fact that electrons don't drift away from the molecule? In other ...
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25 views

Dirac equation in the presence of a defect

The 1D Dirac equation in the presence of a defect is described by a position dependent mass term known as a "kink" or "soliton". It is sign changing and tends to a constant at positive and negative ...
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107 views

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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503 views

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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How do you determine the symmetry of spatial wave functions?

I have been reading about the ways to determine the ground of state of an atom. There are three Hund's rules in determining which electronic state is a ground state. And the second rule says you need ...
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Rectangular potential barrier

Take the usual rectangular potential barrier, that is: $$V(x)=0 \: \text{if} \: x<0 \: \text{or}\: \: x>a$$ $$V(x)=V_0 \: \text{if} \: 0\leq x \leq a.$$ I've looked at several notes and books ...
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How can the wave function contain all information of a system?

During my quantum mechanics lectures and in literature I sometimes hear that "the wave function, $\Psi$, contains all information of the system". This has made me feel rather puzzled so I hope you ...
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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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State of a system in Quantum Mechanics and state vectors

I'm taking a course in Quantum Mechanics and there is something I'm not being able to fully understand. On more elementary courses on Quantum Mechanics I've been told that the idea of Quantum ...
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Need help on understanding mechanical wave function [closed]

My textbook states that, equation 1 : y(x=0,t) = Acos($\omega$t) = Acos(2$\pi$ft), which I understand. However the book goes deeper stating also that, t-$\frac{x}{v}$, and $\frac{x}{v}$-t I am ...