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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Expectation value of $(1/r^2)$ of radial wavefunction [closed]

How is this equal to zero....?
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How to arrive at the Bloch equation $H(k)u(k) = E(k)u(k)$?

Bloch's theorem states that in the presence of a periodic potential solutions to the Schrödinger equation take the following form: $$Ψ(k) = exp(ik•r)*u(k)$$ I am trying to show that using this ansatz ...
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Can I see $Ψ^∗(x,t)$ as a linear functional which can be aplied on wavefunction $\Psi(x,t)$?

Let's say I have a wavefunction $\Psi(x,t) = A e^{i(kx−ωt)}$. Now I complex conjugate it which gives me $\Psi^∗(x,t)$. My first question is: Does $\Psi^∗(x,t)$ live in the dual of a Hilbert space? My ...
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How large should the wavefunction “box” (Schrödinger’s cat in a box) be with respect to the particle dimensions to be an “outside the box observer”?

I wonder what the ratio between particles with an uncollapsed wavefunction and particles with a collapsed wavefunction is. On earth, in a galaxy or in the universe. It seems to depend on the position ...
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How is particle type "encoded" in wave function?

Let single-particle wave functions most generally be given by $\big|\psi\big\rangle\in\mathcal{H}$, where $\mathcal{H}=L^2$ is the space of square-integrable functions. In the first place, the wave ...
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Infinite potential well chapter, To normalize wavefunction problem: apply concept [closed]

Suppose an electron in an infinite potential well with width, L, has a wavefunction, ϕ(z)=Az(z−L) for 0<z<L Normalize this wavefunction and derive an expression for the constant A in terms of L. ...
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Infinite well potential problem: apply concept [closed]

Consider an infinite potential well defined as $V \left( x \right) = 0, ~~ -5nm < x < 5nm$ V(x) = infinity, otherwise Suppose an electron is in the $n = 3$ state in this infinite potential ...
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Why does the Energy of a particle in a Potential Well have to be greater than the minimum value of the Potential? [duplicate]

I was going through Griffiths Intro to Quantum and came across this This implies the following potential situation, would have red as a normalizable solution, and green wouldn't. I understand the ...
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Husimi $Q$-function of Infinite Square well

Eigen-Wavefunction of infinite square well is $$\psi(x)=\sqrt{2/l}\sin(n\pi x/l).$$ I want to write Husimi $Q$ function for infinite square well. General expression of Q function is $$Q=(1/2)\pi \...
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A doubt regarding Quantum Harmonic Oscillator

Classically when we solve Newton's equation for $V=\frac{1}{2}m\omega^2x^2$ we get two linearly independent solutions (for $\omega\not=0$): $Ae^{\omega t}$ & $Be^{-\omega t}$, their linear ...
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What is the ratio between particles with an uncollapsed wavefunction and particles with a collapsed wavefunction? [closed]

I wonder what the ratio between particles with an uncollapsed wavefunction and particles with a collapsed wavefunction is. On earth, in a galaxy or in the universe.
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Can solutions to the time-independent Schrodinger equation give solutions to the general equation if the potential is time dependent?

I am reading through Griffiths Intro to Quantum, and he outlines how one can find solutions to the Schrodinger equation, by assuming the potential is constant in time (no time dependence), thus you ...
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Evolution of a position state in an infinite well potential

Let the potential be $$V = \infty \hspace{3cm}(0>x, x>L)$$ $$V = 0 \hspace{3.7cm}(L>x>0).$$ Now, we measure the position of a particle and discover it is located at $L/4$. What is the ...
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Creation and annihilation operator applied to non-basis vector

Suppose we are given the vector $$|n_1, n_2, \ldots \rangle \in H^{\otimes n}_s$$ where $H^{\otimes n}_s$ is the $n$-fold symmetric tensor product of a Hilbert space $H$, and $|n_1, n_2, \ldots \...
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Confusion on states in level scheme in the popular deuteron isospin and spin explanation

I'm having trouble understanding a specific part of the classic deuteron example when introducing Isospin. I have seen this exact example in multiple lectures and textbooks (recently "Nuclear ...
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What does the Wheeler-DeWitt equation imply about the Schrödinger equation concerning the wave function?

The WDW equation is: $\hat{H}(x)|\psi \rangle=0.$ Schrödinger’s time dependent wave function equation says: $$i\hbar \frac{\mathrm d}{\mathrm dt} | \Psi(t)\rangle=\hat{H}|\Psi(t)\rangle.$$ Does it ...
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Single vs. Double Dirac Potential

Both single and double Dirac potentials have an even ground state solution. Supposing the 'strengths' $\alpha$ of the single and double Dirac potentials to be the same, i.e., supposing $V_{single} = -\...
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What is Wannier state in solid-state physics? [closed]

I only retrieved information about the Wannier function, but I still don't know exactly what the Wannier state is. I would appreciate it if someone could explain it.
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Probability of a state "being found in another" [closed]

I'm sorry if something gets lost in translation, as my professor wrote all questions in portuguese, but what does it mean to ask the probability of finding "state $|\alpha \rangle$ in state $|\...
4 votes
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Assigning initial conditions for Schrodinger's equation

I am self-teaching myself quantum mechanics, and my understanding so far is as follows. In the most general case, we would like to find a wave function $\varphi(x,t) \in \mathcal{H}$, where $\mathcal{...
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Does an electron flying through empty space stay on a fixed trajectory, or does it jitter around randomly?

Electrons exist as a probability cloud defined by their wave function. If an electron was flying through empty space (e.g. as beta radiation), would it (a.) follow a fixed straight path (which we ...
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Doubt regarding the completeness of ${\psi_n}$ in infinite potential well [duplicate]

The wavefunctions (without the time factor) for an infinite potential well (width: $0$ to $a$): $$\psi_n=\sqrt{\frac{2}{a}}\sin\left(\frac{n\pi x}{a} \right).$$ The set of $\psi_n$ is complete as any ...
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Different versions for the Radial Solution of the Hydrogen Atom [closed]

Initially, I'm trying to prove if the following is true $$|\psi_{n00}(r=0)|^2 =\dfrac{1}{\pi n^3a_0^3}$$ I'm looking to solve some calculations made in Introductory Quantum Optics Christopher Gerry ...
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How can the position representation make sense with compatibility of addition? (Dirac Notation)

According to the definition of complex inner product is that: $$⟨\psi|\phi_{1} + \phi_{2}⟩ = \left<\psi|\phi_{1}\right> + \left< \psi| \phi_{2} \right>, \forall \psi, \phi.$$ This implies ...
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What is the difference between the state $| \psi \rangle$ of quantum mechanics and the microscopic state $St(q,p)$ of statistical mechanics?

In terms of physical quantities, the $|\psi \rangle$ of quantum mechanics and the microstate $St(q,p)$ of statistical mechanics are both a vector, and the microstate of statistical mechanics can be ...
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Why are exact solutions limited to hydrogen-like atoms? [duplicate]

Why can we only find exact solutions to the Schrödinger equation for Hydrogen atoms without estimating. What is the problem with the mathematics of extending the Schrödinger equation to more ...
4 votes
2 answers
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Applying measurement postulate to a continuous sum of eigenvectors (by analogy)

Measurement postulate: If we measure the Hermitian operator $\hat Q$ in the state $Ψ$, the possible outcomes for the measurement are the eigenvalues $q_1$, $q_2$, . . .. The probability $p_i$ to ...
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Why is the probability current density $J$ zero when $Ψ$ is purely real or imaginary?

My intuition says that there is both a real and imaginary part to Ψ, so having a purely real/imaginary Ψ cannot work.
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Basic Quantum mechanics: This nonphysical "fun" homework with operators just gives me nonsense [closed]

For QM homework we are given $\vert \psi \rangle = \sum_{i = 1}^3 c_i \vert i \rangle$ where $\vert i \rangle$ represent different positions. Firstly we are asked $P(i=2\vert \psi)$ which is given $P(...
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Which experiments confirm that the Born rule is $|\psi|^2$ rather than $|\psi|$?

It seems like some experiments on quantum systems, like the electron $g-2$ measurement, do not rely directly on the Born rule, since they are more so measuring inherent characteristics of the ...
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How can we understand the $g$-factor as a property of the wavefunction?

I saw some fancy derivation of the fact that $g = 2$ in the Dirac equation, but I couldn't really follow it. And I imagine it's about 100 times harder in QED. What I really want is to understand ...
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Why do we suppose that reflected quantum state has the same energy that initial state?

I just read in my lecture notes that the teacher assumed the wave vector $k_+$ of reflected (from potential) state the same (but inverted) as wave vector of initial psi function (that goes to the ...
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Doubt tensor product states of bipartite system

Consider a system $A$ whose basis states are $|\phi_1\rangle_{(i)}\in H^{(1)}$ and a system $B$ whose basis states are $|\phi_2\rangle_{(j)}\in H^{(2)}$. Then the basis states of the combined system ...
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Justifying separation of variables in solving the Schrödinger equation in 3D

Consider a 3D infinite square "box", satisfying the time-independent S.E. $$ \begin{align*} -\frac{\hbar^2}{2m}\nabla^2 \psi &= E\psi, \;\;\; x,y,z\in[0,a], \\ \psi &= 0, \;\;\;\;\;\...
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The intepretation and maths behind the many-worlds interpretation of quantum mechanics

recently I started reading the book "Something deeply hidden" by Sean Caroll. In the book he talks about the many-worlds interpretation of quantum mechanics as a more elegant way of thinking ...
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„Orbital“ of an quark

Inspired by the idea of the electron orbitals ( probability of finding an electron in an atom) i was wondering what that would look like inside a proton or neutron for quarks. For simplicity consider ...
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Wavefunction with two different values at same point

Consider a particle on sphere. Its Hamiltonian in spherical polar coordinates is given by - $-\frac{\hbar^2}{2mr^2}\Big(\frac{1}{\sin\theta}\frac{\partial}{\partial\theta}\sin\theta\frac{\partial}{\...
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Boundary condition like $\psi(0)= \psi(L) = 0$ [duplicate]

This might be very elementary. But I have been baffled for a while. For the infinite square well potential, the boundary condition is that $\psi(0) =\psi(L) = 0 $. However, from real analysis, we know ...
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Not quantum field theory, but rather ordinary quantum mechanics many-body wave function

Reading standard texts of quantum mechanics as such, as distinct from quantum field theory, I am left not knowing the principle that defines a wave function for a closed quantum system constituted by ...
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Eigenvalue problem of $L_z$

From Shankar's QM book pg. 313, the eigenvalue problem for $L_z=XP_y-YP_x$ in polar coordinates is $$-i\hbar \frac{\partial \psi(\rho,\phi)}{\partial \phi}=l_z\psi(\rho,\phi)$$ since $L_z=-i\hbar\frac{...
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Principled definition of many particle wave function

In standard texts, I find no systematic and principled definition of a many-particle wave function. Perhaps I am not looking in the right standard textbooks. In my inadequate reading of the literature,...
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How to prove this equation?

In J.J. Sakurai's Modern QM(Third edition), section 6.4.1, the author says that the equation 6.104 $$ \frac{e^{i\bf{k\cdot x}}}{(2π)^{\frac{3}{2}}}=\frac{1}{(2π)^{\frac{3}{2}}}\sum_l(2l+1)i^lj_l(kr)...
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In the electron double slit experiment, what is interfering?

According to quantum field theory, an electron particle is an excitation of the electron field. Is it the waves of excitation in the electron field that are interfering in a double slit experiment? ...
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1 answer
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Wave function Fourier transform with time

I found the Fourier transform at $t=0$ for the wave function of a wave packet (and it's inverse Fourier transform) : $$\Psi(x,0)=\frac{1}{\sqrt{2\pi}}\int_{-\infty}^{\infty}\Phi(k)e^{ikx}dk$$ $$\Phi(k)...
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Calculating $\langle p^2 \rangle $ for a non-smooth wavefunction [closed]

I am trying to calculate $\langle p^2 \rangle$ for the wavefunction $$ \psi(x) = \frac{\sqrt{m\alpha}}{\hbar}e^{-\frac{m\alpha\lvert x \rvert}{\hbar^2}}.$$ So far I have $$ \langle p^2 \rangle = - \...
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What is the difference between Dirac delta function orthogonality and Kronecker delta orthogonality?

In the derivation of Bloch Wave, I encountered a problem. First of all this is the definition of Bloch Wave: $$ \psi_{n\mathbf{k}} (\mathbf{r} ) = e^{i\mathbf{k} \cdot \mathbf{r} } u_{n\mathbf{k}} (\...
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How does branching from the many-worlds interpretation come from the assumption of a universal wavefunction?

In the many-worlds interpretation, the cut between the observed and the observer, the quantum and the classical is removed, and it logically follows that there can be a universal wavefunction. When ...
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Why isn't the delta-function-potential solution the same as the infinite square well solution?

In Griffiths, there is a worked-through derivation for the solutions to the wave function for a delta-square potential case and for the infinite square well case. The infinite square well solution ...
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Interpretation of wavefunction in the Feynman formulation of QFT

Feynman had this alternative way of doing QFT in which particle states are evolved forward in proper time. Specifically, the propagator is given by : $$U(x_1,t_1,x_2,t_2)=\int _0^{\infty} d\tau \...
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The connection between the Heisenberg Uncertainty Principle and wavefunctions

I have recently read Something Deeply Hidden by Sean Carroll, and in the book, he tries to sum up how the Heisenberg Uncertainty Principle can be found from just the way wavefunctions work, but I ...
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