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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189 views

How can quantum wavefunctions be smooth/continuous when particles are created/destroyed/changed?

My (admittedly limited) understanding of the Schrodinger equation tells me that the vector differential operators are only meaningful over a differentiable phase space. For example, if the dimensions ...
3
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2answers
27k views

What is the difference between the Bohr model of the atom and Schrödinger's model?

What is the difference between the Bohr model of the atom and The solution of the Schrödinger equation for the hydrogen atom? Are there any difference between definition of the electric potential ...
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0answers
48 views

Wave packets in Dirac equation

Gaussian wave packets remain Gaussians after evolution in case of the Schrodinger equation. It is a very useful property of these wave packets. I don't think the same is true for a Gaussian wave ...
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2answers
514 views

Floquet quasienergy spectrum, continuous or discrete?

I haven't got a feeling about Floquet quasienergy, although it is talked by many people these days. Floquet theorem: Consider a Hamiltonian which is time periodic $H(t)=H(t+\tau)$. The ...
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1answer
105 views

Derivation of group velocity using Fourier transform

The aim is to determine the group velocity of a wave packet with the general form $$\Psi\left(x,t\right)=\frac{1}{\sqrt{2\pi}}\int_{-\infty}^{\infty} \phi\left(x\right)e^{i\left(kx-\omega t\right)}dk....
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25 views

Why can a free particle not exists in a stationary state? [duplicate]

In normalising the complete wavefunction of a free partcile; $V(x)=0$ we arrive at $\int_{-\infty}^{\infty}\Psi_{k} ^\dagger\Psi_{k}dx=|A|^{2}\int_{-\infty}^{\infty}dx=|A|^{2}\left(\infty\right)$ ...
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0answers
51 views

How can we fix the constant of the energy eigenstates of a quantum free particle such that they satisfy the orthonormality condition?

For a quantum free particle, the momentum and energy eigenstates are compatible. The constants of the momentum eigenstates are fixed by their orthonormality. Similarly, how can we fix the constant for ...
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0answers
22 views

For a quantum free particle, would it be possible to relate the wavefunction $a_E(E)$ in energy basis and $a_p(p)$ in momentum basis?

The energy for a free particle has continuous energy eigenvalues $E$. Let $u(E,x)$ be its energy eigenstates in position basis. Its wavefunction $\psi(x,t)$ can be expressed as \begin{align} \psi(x,t)...
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0answers
52 views

In quantum descriptions of atoms why are observables (which we derive from the wave function) attributed to electrons?

For example the orbital angular momentum, for the hydrogen atom. Is this the total angular momentum of the atom(electron and proton) or just the electron? I am asking because, I am learning about how ...
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3answers
88 views
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4answers
302 views

Why does the electron wave function collapse in a double slit experiment?

Did the electron wave function collapse in the double slit experiment due to being observed, OR is it that the electron wave function collapsed because the instrument used to measure it physically ...
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1answer
78 views

Physical interpretation of the constant coefficient appearing in solution to the Schrodinger equation

The product solution to the Schrodinger's equation is $$\Psi_{n} \left ( x,t \right )=\psi\left ( x \right )\phi\left ( t \right )$$ By superposition, the solution becomes $$\Psi \left ( x,t \...
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2answers
240 views

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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1answer
55 views

Quantum Backaction Question

Question about quantum back action in hypothetical scenario: We know that, at $t_0$, a certain kind of particle, with spin initially prepared to be “spin right” in the x basis, goes through a Stern-...
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0answers
61 views

Infinite square well physical interpretation

In quantum mechanics, the description of the infinite square well is given with the potential energy defined as $$V(x) = \begin{cases} 0 & \text{if } 0 \leq x \leq a,\\ \infty & \...
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1answer
102 views

Finite two-dimensional potential square well [closed]

I'm trying to solve the Schrodinger equation $$ -\frac{\hbar^2}{2m}\nabla^2 \psi+V(x,y)\psi(x,y)=E\psi(x,y) \tag{1}$$ for the finite two dimensional potential square well, that is, where $$V(x,y)=\...
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0answers
31 views

Probability of measuring an observable $P$ in state $f$, computation [duplicate]

I have state vector $$f(x)=e^{-|x|+ix}$$ and observable $$P=-i\frac {d} {dx} $$ probability that measurement of $P$ in state $f$ will be in $[-1,1]$ I am stuck on this step. I dont know how to take ...
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1answer
44 views

Energy Expectation Value

I had an assignment question in which I was asked to calculate the expectation value of energy, $\langle E\rangle (t),$ and in the solution to it, the following was stated: \begin{align*} \langle E\...
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2answers
64 views

Transmission and reflection amplitudes for delta potential Schrodinger equation

I hope this question is not too straightforward for this Q&A site. I have been reading a set of notes in which the transmission and reflection amplitudes for the delta potential Schrodinger ...
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1answer
43 views

Why is the wave function an element of the function space? [closed]

The general wave function is of the form $$\Psi \left ( x,y,z,t \right )=\psi \left ( x,y,z \right )T\left ( t \right )$$ Solving via separation of variables and finding the product solutions yields,...
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1answer
335 views

How is it possible to pull out derivatives of a wavefunction?

In an early derivation, the following equation was stated: $$\frac\partial{\partial t}\lvert\psi\rvert^2 = \frac{i\hbar}{2m}\biggl(\psi^*\frac{\partial^2\psi}{\partial x^2} - \frac{\partial^2\psi^*}{\...
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4answers
141 views

Is there a reason why probability density is written as $\psi^*\psi$ instead of $\psi\psi^*$?

As the title states, I see $|\psi|^2$ written as $\psi^*\psi$ instead of $\psi\psi^*$. Are both correct or is there a reason behind it? As far as I'm aware, the only time I see this sort of ordering ...
2
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1answer
222 views

Analogy to Fourier transform in spherical coordinates with boundary at a certain radius

Suppose, we have a wavefuction $\phi(\vec{x})$ which is restricted in a sphere, with the spherical boundary condtion $$\phi(\vec{x}=R)=\phi_0.$$ How can I do the 'Fourier transformation' as the case ...
3
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1answer
90 views

How do we find the number of bounded states in this potential?

for the potential $$V(x)=-\frac{1}{1+\frac{x^2}{m^2}}$$ we can approximate the wave function and bounded state accurately for $x << m$ as simple harmonic oscillator, so what are we gonna do if ...
3
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1answer
90 views

Do wave functions really belong to $L^2$ space, or do we need to restrict our physical Hilbert space even further?

I am beginning to study quantum mechanics and I got stuck right at the beginning. I am trying to prove that the time derivative of the expected value of momentum of a particle is the (negative) ...
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0answers
112 views

Guess the wave function in a given potential

Are there any techniques in guessing the ground state wave function in any given potential? For example, for a given potential like $$ \frac{1}{1-x^2}$$ or $$ \frac{1}{1-x^3}~?$$ I know wave ...
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2answers
91 views

Measurement of energy apparently violating the position-momentum Uncertainty Principle in a potential that does not depend on distance?

I am taking a beginning course in QM and I have learnt that the measurement of energy collapses the wavefunction of a particle to one of its energy eigenstates. But some misconceptions regarding this ...
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2answers
117 views

Normalisation of free particle wavefunction

The wavefunction $\Psi(x,t)$ for a free particle is given by $$\Psi(x,t) = A e^{i(kx-\frac{\hbar k}{2m}t)}$$ This wavefunction is non-normalisable. Does this mean that free particles do not exist in ...
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0answers
47 views

Additional quantum states of the infinite square well

The quantum states $\psi(x)$ of the infinite square well of width $a$ are given by $$\psi(x) = \sqrt{\frac{2}{a}}\sin\Big(\frac{n \pi x}{a}\Big),\ n= 1,2,3, \dots$$ Now, I understand $n \neq 0$, as ...
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3answers
103 views

Modern explanation of the Young experiment with Quantum Field Theory?

In the Young double slit experiment it is possible to detect the arrival of individual photons as well as an interference pattern. It doesn't makes much sense to me that something could be either a ...
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1answer
97 views

Bound states of Dirac Delta function in infinite well

If there is a potential of $-\alpha\delta(x)$ for $-a<x<a$ and $\infty$ elsewhere, and the energy of the system is less than 0, then I'm trying to find the wave function. From the Schrodinger ...
2
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0answers
32 views

How come an electron's wave function being nonzero at far distances doesn't mean it can travel faster than light? [duplicate]

I think the wave function of a free electron is nonzero almost everywhere. In particular there are regions of space arbitrarily far away where the electron has positive probability of being found. If ...
3
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1answer
88 views

What happens to the wave function of a particle immediately after measuring its energy?

For this question, I will be adhering to the Copenhagen interpretation (since that's what I've learned in university so far). For the sake of brevity/clarity, also, assume the Hamiltonian here has ...
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5answers
378 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} x^{_{...
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1answer
56 views

Antisymmetry requirement for the total wavefunction

My understanding is that if we are dealing with a system of two electrons, the total wavefunction needs to be antisymmetric only when the two electrons have same value of n and l ( i.e. they are ...
21
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2answers
779 views

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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2answers
56 views

Measurement of position after collapse of a wavefunction

Suppose I have a wavefunction which collapses to a certain eigenstate after a measurement of energy. In that state, I perform a calculation of position and obtain a certain position value, say $x_0$. ...
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4answers
849 views

How does wave function collapse when I measure position?

Text books say that when you measure a particle's position, its wave function collapses to one eigenstate, which is a delta function at that location. I'm confused here. A measurement always have ...
1
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0answers
43 views

Showing two wavefunctions are proportional to one another [duplicate]

I am struggling to answer the following question: Let ψ₁(x) and ψ₂(x) be normalisable energy eigenfunctions for a particle of mass m in one dimension moving in a potential V(x). Suppose that ψ₁ and ...
2
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2answers
2k views

Matter waves and de Broglie wave length

The wavelength of a particle of momentum p is calculated using De Broglie relation. The de Broglie relation was postulated for what is called a matter waves. Now according to the statistical ...
2
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2answers
126 views

How do we decide whether an electron orbital has a non-zero or zero probability of lying inside the nucleus of an hydrogen atom?

How do we decide whether an electron orbital has a non-zero or zero probability of lying inside the nucleus of an hydrogen atom? It is mostly from the radial function, as to what I think but how ...
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1answer
113 views

Schrodinger equation violates mathematics?

By the Hamiltonian formalism of quantum mechanics, given a quantum system in a state $\Psi$ in a Hilbert space $\mathcal H$, the state will instantaneously evolve in time according to $$\dot{\Psi}=\...
3
votes
1answer
182 views

Continuity behaviour of a wavefunction when the potential exhibits a discontinuity

If a potential $V(x,t)$ exhibits a finite discontinuity in space, the wavefunction $\phi(x,t)$ and its spatial derivative will be continuous. If a potential exhibits a finite discontinuity in time, ...
0
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2answers
288 views

Time dependent and time independent Schrödinger equations

I'm trying to understand the relation between the time dependent and time dependent Schrödinger equations. In particular, we know that the TDSE is $$H\Psi=i\hbar \frac{\partial \Psi}{\partial t}$$ ...
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53 views

solutions of wave equation with cubic term

Does the following equation $$ \nabla^\mu \nabla_\mu \psi + a \psi^3 = b \psi $$ where $\psi$ is a real function, $a$ and $b$ are real constants, have other solutions that extend beyond a one ...
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1answer
100 views

Infinite potential well question with wave function $\psi (x) = (x -a/2)^2$

In an infinite potential well with width $a$, a particle in this potential well is at state with wave function is $\psi (x) = (x -a/2)^2$ (not normalized). If you measure the energy of the ...
0
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3answers
93 views

Quick way to compute $\langle n^{'}l^{'}m^{'}|r^k|nlm \rangle$, $k \in I$; $|nlm\rangle$ is $H$ atom eigenfunction [closed]

I want to compute quickly (using maybe some scaling arguments) $\langle n^{'}l^{'}m^{'}|r^k|nlm\rangle$, where $k \in I$. $|nlm \rangle$ is the eigenfunction of the Hydrogen atom ($H$). Example: ...
0
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1answer
85 views

Confused on how to interpret the energy eigenfunction of Hydrogen

So here is an image of the third lowest energy eigenfunction of an electron in a hydrogen atom: Image from http://imgur.com/Lu4MocL I understand well the eigenfunctions given by Schrodinger's ...
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1answer
54 views

Spinor expectation value and measurement

I have a question about the difference between expectation value and probability of measurement. consider the spinor $\zeta = [-3\ \ 4i\ ]^T$ . The expectation value of $S_x$ is zero because : $$\...
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2answers
144 views

Coefficients and wavefunction in quantum mechanics

In general quantum mechanics we represent the state of a system with a state vector $| \psi \rangle $ in some Hilbert space in some base. Assuming a complete discrete set of bases vectors $ |n \rangle ...