Questions tagged [schroedinger-equation]

Partial differential equation which describes the time evolution of the wavefunction of a quantum system. It is one of the first and most fundamental equations of quantum mechanics.

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Wave packets not satisfying the Schrödinger equation?

The time-independent Schrödinger equation of a free particle in 1 dimension is $$ \begin{equation} -\frac{\hbar^2}{2m}\partial^2_x\psi(x) = E\psi(x) \end{equation} $$ which has solutions in form of $...
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Proving that the sum of an operator and it's adjoint is Hermitian

I can solve the two questions using matrices in math, but we are expected to show it in form of the property of Hermitian operators:$$\int_{-\infty}^{\infty}\Psi^*O\Psi=\int_{-\infty}^{\infty}(O\Psi)^*...
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Quantum Mechanics And Cauchy-Riemann Equations

A complex valued function in complex variable $z$ can be used to represent the potential and field lines of the electric field in 2 dimensions: Some notes on the method of Conjugate Functions This is ...
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Eigenstates in a 2D metal disk with finite-size perpendicular magnetic field (Piece-wise Gauge)

Let's have a two-dimensional metal disk with radius R, and now I apply a uniform magnetic field B perpendicular to the disk, within the concentric part of the disk with radius r $\lt$ R. How can we ...
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26 views

Property of the time evolution operator preserving the norm of the wavefunction

Since the time evolution unitary operator preserves norm, if applied to any system say electron whizzing around its orbitals, no matter what time we consider would it always have the same probability ...
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61 views

Schrodinger equation vs Feynman diagrams

If one wants to assess how an electron orbits in a hydrogen atom one uses the Schrodinger equation. Ditto for an electron in a magnetic well. However if one wants to assess how particles interact or ...
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The particle in a box problem

I'm studying undergrad level chemistry with no strong background in physics. So the problem is a little confusing to me. A few questions for clarification: Using the electron on a 2-dimensional box ...
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60 views

Why does the choice of origin affect the wave function?

Consider a particle in a 1D-Box. The box ranges from x=0 to x=l in the first case, and from x=-l/2 to x=l/2 in the second case. The only difference I see is that the origin is shifted. On solving ...
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Scale invariance in (2+1)D nonrelativistic field theory

Context: I am reading a paper named 'Nonrelativistic field-theoretic scale anomaly' on scale invariance in nonrelativistic field theory. The Lagrangian density for the scalar field is given by, $$\...
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59 views

Differentiability of wave function at boundary in infinite square well

I was told in class that a wave function should have the following properties: Finite and single-valued Continuous Differentiable Square integrable But if we consider the wave function in an ...
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Question on modified particle in a box [on hold]

If say, I have a particle bound in an infinite well where the floor is very slightly sloping. How would I sketch a possible wave function when the energy is E1(no nodes)? How do we do this ...
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Is there an accepted way to plot a wave function and the potential?

When using the time-independent Schrodinger equation and finding a wave function $\psi(x)$ for a given potential $V(x)$ is there a consistent way to plot these two objects on the same image that takes ...
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Does Hatfield give the correct 1st excited state for a scalar field's wave-functional?

Hatfield's QFT textbook gives a scalar field $\phi(\vec{x})$'s first excited state as $$ \Psi_1[\tilde\phi] = \left(\frac{2 \omega_{k_1}}{(2\pi)^3}\right)^{\frac 1 2} \tilde\phi(\vec{k}_1)\Psi_0[\...
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61 views

Can one find the uncertainty of a particle from Schrödinger's wave equation? [closed]

Right now I am studying quantum mechanics and I'm having trouble understanding what exactly $\Psi$ is in Schrodinger's equation $\Psi(x) = A\sin(kx) + B\cos(kx)$. After doing some googling I learned ...
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What is evolving with time?

In Griffiths, we are told that the expansion coefficients of the stationary states are simply complex numbers: $$\Psi(x, \ t) \ = \ \displaystyle\sum_{n} c_n e^{-iEt/\hbar} \psi_n(x)$$ How do we ...
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Has anyone solved the inverse hydrogen atom problem?

This question was motivated by a question asked yesterday (How to find the electrostatic potential of a hydrogen-like charge density?). This got me wondering about what would result from solving the ...
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Differences betwen the conformal group and the Schrödinger group?

Facts: The Maxwell (free) equations (4d) are invariant under the 15 dimensional conformal group. The free Schrödinger equation in 3d is invariant under the 15 dimensional group "called" Schrödinger ...
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Double well potential and choosing coefficients

During lecture on single potential well, my professor had the following wavefunctions for the three regions: $\psi(x) = e^{\kappa x}\ for \ x<-L$ $\psi(x) = B\cos(ikx)$ for $-L<x<L$ $\...
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Does the “Particle in a box” mean space & motion are quantized?

Recently I had a conversation with someone about quantum mechanics, I was asking if it meant everything was quantised. If space and our ability to move through it and the positions matter could take ...
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Why can we use single electron Schrodinger equations to describe electrons in solids?

A solid is clearly a many electron system. Yet we often use single electron Schrodinger equations to calculate the quantities of interest. Probably this is most common in semiconductors. Why is that, ...
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Energy or a piecewise defined wave function in an Infinite Square well

So I'm asked to determine the most probable measured energy value of a particle in an infinite square well with wave function $$\psi(x)=\begin{cases} Ax, & 0< x<\frac{a}{4} \\[1em] ...
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Does the second derivative of wave function have to be continuous? [duplicate]

I am very new to quantum mechanics and I have a question about wave function in time-independent schrodinger equation. In Beiser modern physic, they say the first derivative of wave function should be ...
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Question on some basic principles of quantum theory [closed]

I asked a physicist (former head of the physics department in a university) about some of the basics of quantum theory and the double slit experiment. Here is his reply. Are any of these points ...
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59 views

Solve time-independent Schrödinger equation in the momentum basis

So I was reading Henk Stoof's Ultracold Quantum Physics and there was this simple example of a particle in a finite space (not infinite well) with no potential whatsoever. It is solved in the position ...
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Proof of Bloch's Theorem

I'm reading the book Solid State Physics by Ashcroft and Mermin. In its second proof of Bloch's Theorem on p.137, the periodic potential $U(\mathbf{r})$ and the wavefunction $\psi(\mathbf{r})$ both ...
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61 views

Scattering states for even potential in 1D

E.g. For a finite square well that has the following potential: $$ V(x)= \begin{cases} 0, & |x|>a \\ -V_0, &|x|\leq a ...
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Is there an analog of the Kirchhoff integral theorem applied to the Schrodinger equation?

The Kirchhoff integral formula is a powerful tool that allows us to compute solutions to the standard wave equation given certain 2D boundary conditions. Is there anything similar that holds for the ...
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General solution of a particle in a box

The general solution of particle in a 1D infinite potential well(width L) is given by: $$\psi(x,t)=\sum_n a_{n}.\sqrt{\frac{2}{L}}.\sin\bigg(\frac{n\pi x}{L}\bigg).\exp\bigg(-\frac{ iE_n.t}{\hbar}\...
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How does $\psi(x) =\exp(\frac{iq}{\hbar}\int^xA(x')\cdot dx')\phi(x)$ remove the gauge field for a free particle?

In what sense does writing $$\psi(x) =\exp(\frac{iq}{\hbar}\int^xA(x')\cdot dx')\phi(x)$$ "formally remove the gauge field" for a free particle in the Hamiltonian $$H\psi=\frac{1}{2m}(-i\hbar\nabla -...
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Why are the left- and right-hand sides of a differential equation with two separated variables equal to a constant?

While deriving the Time Independent Schrodinger Equation, my book mentioned this line. So time and position of a particle are two independent variables. If they are equal to one another for all ...
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Does the ground state of the Schrödinger equation, in any number of dimensions, always have constant phase?

I just read this argument in this paper (PDF). It suggests that, from variational principles, you can show that you can always lower the energy of a state by making the phase constant, thus resulting ...
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138 views

Connection between Schrödinger equations for a finite triangular well and a finite square well

Suppose we solve the Schrödinger equation, \begin{equation} -\psi''(x) + V(x) \psi(x) = -|E| \psi(x), \end{equation} where two cases are considered, $V(x) = -V_0$ (square well) and $V(x) = -V_0 (1 - |...
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What is the quantum mechanical turning point of the $n^{th}$ energy eigenstate of an oscillator? [closed]

I am looking for an analytical expression for the most likely position for a quantum harmonic oscillator (which I refer to as the quantum mechanical "turning points"), in terms of $n$. For the quantum ...
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84 views

Linearity of Schrödinger equation and perturbation theory

So, I was studying quantum mechanics and reached the point where perturbation theory is discussed. It is my first time in this topic, and something called my attention: it was said that we need ...
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120 views

Analogies between equations [closed]

What properties of fields and matter are related to the analogy of the Schrödinger equation and the Navier-Stokes equation, between the equation of general relativity and the Navier-Stokes equation? I ...
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One-dimensional Schrödinger equation: reproducing a given set of energy values [duplicate]

Given a set of $N$ increasing real numbers $\{E_1, E_n, \cdots, E_N \}$, is it always possible to find a potential $V(x)$ such that the set of $\{E_j\}$ are the lowest eigenvalues of the corresponding ...
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What's the intuition for the reflection of a quantum particle at a potential step equal to the particle's energy?

While doing the problem of potential step, I saw that if the energy of the particle is equal to the potential energy of the step, then the wave function is a constant, or to say the probability ...
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Odd potentials in TISE

When we have an even potential we say that it has an even and odd parity wavefunctions, cf. e.g. this & this Phys.SE posts. What about an odd potential? For example, two delta functions centered ...
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71 views

Show Galilean invariance of Schrödinger eq

I'm trying to show the Galilean invariance of the (time-dependent) Schrödinger equation by transforming as follows: $$ \left\{\begin{eqnarray}\psi(\vec{r},t) &=& \psi(\vec{r}'-\vec{v}t,t),\\ \...
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59 views

Central potential vs non central potential (particle in a box)

Whenever we are given a central force potential, we keen to investigate how the energy levels are related to the Angular Momentum operator like $L_z$, $L^2$ etc. And definitely they commute with the ...
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Are stationary states always real functions? [duplicate]

I've noticed that for many quantum potentials, (the harmonic oscillator, the infinite square well, and the delta potential) the wavefunctions $\psi_n$ of stationary states are always real valued ...
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Why can't the wave function and it's derivative be zero at the same point? [duplicate]

This was discussed before proving the node theorem.
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Confusion regarding the bound state of a Delta-function potential and Tunneling

I was reading (Griffith's QM book) about the Bound states for delta-function potential of the form $-\alpha \, \delta(x)$ where $\alpha > 0$. I feel a bit conceptually unclear. Few doubts I have ...
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Interaction picture Sakurai

I’m going through Sakurai and got stuck with the following in the interaction picture subsection $$i \hbar \frac{\partial}{\partial t}\left|\alpha, t_{0} ; t\right\rangle_{I}=i \hbar \frac{\partial}{\...
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117 views

Which wave function should I adopt?

Suppose I have Hamiltonian $H_0(\hat{p},\hat{r})$, it satisfies $H_0\psi(p,r)=E(p)\psi(p,r)$. If I make a change from $\hat{p}\to\hat{p}+p_0$, what is the form of the wave function of the Hamiltonian $...
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A Schrödinger Love Poem? [closed]

This question is for those who can see the romantic side of the Schrödinger equations... My GF is studying quantum mechanics, and I was thinking to write it as an expression of a two-particle ...
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61 views

Solution of Time-dependent Schrodinger Equation for Unitary Operator

While reading Quantum Mechanics Book by Sakurai, I found the time-dependent Schrodinger equation for Unitary Operator. $$i\hbar \frac{\partial}{\partial t}\mathcal{U}(t,t_0)=H\mathcal{U}(t,t_0).$$ ...
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Wavefunction of particle in power law potential of type $x^a$?

How could we calculate the wave function of a particle under a potential of form $V(x)=x^a$? Is there any analytical solution or any general feature of such solution (like its an exponential ...
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65 views

${}$ Dirac delta potential

We know that the number of bound states for an attractive delta potential is one. If so what will the number of bound states for a particle in a repulsive delta potential? If $V(x)= +a \cdot \delta(x)$...
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85 views

Energy eigenvalue with Potential $-e^2/ x$ [closed]

If I have potential which are very well-known like, square barrier, or square well, or step potential, What I do is to set the boundary conditions in Schrödinger's equations. Sometime, the ground ...