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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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QM1223344556666664545 [on hold]

Schrödinger's electron electric wave replaced with an electron probability wave but an electron position probability can only represent a positive value or zero and cannot depict a negative value that ...
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1answer
60 views

Schrödinger Equation with Imaginary Potential

I am trying to solve the following equation (in 1D) and stuck in the middle of the way. Here's the equation: $$i\frac{\partial\psi}{\partial t}=C\cdot\frac{\partial^{2}\psi}{\partial x^{2}}+iD\cdot\...
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33 views

Time-independent Schrödinger equation Lagrangian derivation

Recently I was taking a calculus of variations class and our professor casually obtained the time-independent Schrödinger equation for a free particle from the integral (constants dropped) and it's ...
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1answer
65 views

Experimental tests of Schrodinger evolution, position distribution, in square well and other simple systems?

Have the energy eigenfunctions in position space ever been experimentally tested for the simplest system undergraduates encounter when learning quantum mechanics, the square well? If not, what is the ...
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2answers
51 views

What is the simplest possible Hamiltonian that yields an Antisymmetric Wavefunction?

I am using a Split-Operator Fourier Transform (SOFT) technique to solve the time-dependent electronic Schrödinger Equation (TDSE) for a Hydrogen molecule under the Born-Oppenheimer approximation. So I ...
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1answer
139 views

What does $|$ mean in the Schrödinger Equation?

I saw the $|$ symbol in the Schrödinger Equation $$i\hbar\frac{\partial}{\partial{t}}|\Psi(r,t)\rangle=\hat{H}|\Psi(r,t)\rangle$$ But I don't know what the $|$ means. What does $|$ mean in the ...
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143 views

If one puts a delta-function spike inside an infinite square well, is the resulting potential analytically solvable?

It was recently floated in chat that a particle in a box with a delta-function spike inside it, with hamiltonian $$ H = -\frac{\hbar^2}{2m} \frac{\partial^2}{\partial x^2} + V_0\delta(x-a) $$ and with ...
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1answer
75 views

Particle in a box plus step (ground state)

I am trying to come up with a QM problem that: Can be solved analytically Contains a potential that is a sum of some analytically solvable potential and another contribution: $V'=V_0 + V$ Is then ...
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1answer
76 views

Schroedinger equation in Differential geometric language

I have reading about manifolds and tangents spaces and lie derivatives. I have been wondering is there is a way to write Schrödinger equation in this formalism?
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Confused about linearity of wave equation

Currently in a QM class. My understanding is as follows: the schrodinger equation specifies solutions to your system. When we solve it we get a set of wave functions, which form a basis over the ...
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1answer
72 views

Schrodinger's Equation in three dimensions

Consider Schrödinger's Equation, $$H=\sum^3_{i=1} \frac{p^2_i}{2m_i}+V(x_1,x_2,x_3).$$ In one dimensional case, we can analyse the shape of the potential, i.e $$V(x)=\frac{1}{2}m_1 \omega^2_1 x^2$$ ...
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Conceptual understanding of the Quantum Harmonic oscillator

First: When we consider a quantum particle in a harmonic (quadratic) potential we say that this particle is a harmonic oscillator, because it behaves like one. Is this correct? Now let us assume our ...
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53 views

$\delta$ potential has highest probability for highest potential

I can't understand this intuitively. Figure 2.9 in Griffith's QM says that the wavefunction at $x=0$ for a delta function potential is $\sqrt{\kappa}$, and to the right it decays like $\psi_+=\sqrt{\...
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4answers
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Why do wavefunctions for stationary states include $e^{-iEt/\hbar}$? [duplicate]

Stationary states are separable solutions with $\Psi(x, t)=\psi(x)e^{-iEt/\hbar}$. But why is that there? Griffiths (Section 2.1 Stationary states, equation 2.8) says that observables for these states ...
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1answer
53 views

Energy formula for finite potential well [closed]

The energy formula for infinite potential well is $$E=\frac{n^2h^2}{8ma^2},$$where $m$ is the mass of the particle, $a$ is the width of the well but in the case of finite potential well, I actually ...
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'Show that $E$ must exceed $V(x)$ for every normalisable solution of the TISE'. Surely you can show that this isn't true using exponentials?

In Griffith's this is a question, there's the clear answer which jumps out immediately: $$\frac{d^2\psi}{dx^2} = \frac{2m}{\hbar^2}(V(x)-E)\psi.$$ That is, if E is less than V the second deriviative ...
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1D Infinite Square Well: Box Suddenly Increase in Size. Why the coefficients can be derived by inner producting the wave functions?

I have read the question 1D Infinite Square Well: Box Suddenly Increase in Size. How treat this?. There are two key equations in the selected answer. $$\int_0^{2L} dx \ \psi^*_m\left(x\right) \Psi\...
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1D Potential Barrier Boundary Conditions (tunneling)

I'm trying to solve the boundary conditions of a 1D potential barrier for a free particle. The boundary conditions on the continuity of the wave function and its first derivatives leads to four ...
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2answers
110 views

Symmetries of a differential equation, its solutions and hydrogen atom

A symmetry of a differential equation need not be shared by its solutions. However, under that symmetry, the one solution goes to another. For example, consider the time-independent Schrodinger ...
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37 views

QM limit of QFT in Schwartz [duplicate]

In Matthew Schwartz's QFT text, he derives the Schrodinger Equation in the low-energy limit. I got lost on one of the steps. First he mentions that $$ \Psi (x) = <x| \Psi>,\tag{2.83}$$ ...
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1answer
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Infinite vs Finite dimensional Hilbert space

Let us consider an electron in an infinite square well. As we know that the electron has a spin=$1/2$ . The spin operator ($z$-direction) has two eigenvectors which span the vector space. But if we ...
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Green's function for infinite square well

The Green's function can be given in terms of left and right solutions. $G(x,x';k) = \frac{1}{W}\left(\Psi_{L}(x_{<})\Psi_{R}(x_{>})\right)$ But I don't understand how to determine these left ...
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1answer
66 views

Arriving at the Quantum Mechanial Potential From The Energy Eigenvalues [duplicate]

In Quantum Mechanics, we know that given a potential we can solve the eigen value problem to find out the energy eigen values and eigen functions. Now suppose in an experiment we have information only ...
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1answer
42 views

Why we don't need to normalize the scattering states?

I am new to QM, have find some wavefunction in different potentials, but there we need to normalize the wave function, for a reason that - particle should be found somewhere . So a wave-function, to ...
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2answers
68 views

Continuity of wave function derivative

A particle is defined by a wave function, $Be^{-2x}$ for $x<0$ and $Ce^{4x}$ for $x>0$. For the wave function to be continuous at $x=0$, $B=C$. A wave function must be continuous for it to be ...
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86 views

Can Quantum Mechanical Potential have a Probability Distribution

I am currently in my second semester of undergraduate quantum mechanics. We have recently starting discussing two particle systems, usually in relation to spin interactions. In all of our calculations,...
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67 views

Constructing a staggered translation operator

Consider a matrix as a function of position $x$ $$C=\begin{bmatrix} 0 & A(x) \\ B(x) & 0 \end{bmatrix} .$$ Is it possible to construct a matrix $S$ that translates $A$ and $B$ oppositely? $$...
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1answer
82 views

Schrödinger Equation for a freely falling body near the surface of Earth

Near Earth's surface the Schrödinger equation of a freely falling particle takes the form, $$ \frac {-\hbar^2}{2m} \frac {d^2 \psi (y)}{dy^2} + mgy\psi (y) = E \psi (y). $$ Putting $k=\frac {\sqrt {...
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‘Supersymmetrizing’ an arbitrary quantum-mechanical potential

To my understanding, it is not possible to $``\text{supersymmetrize}"$ an arbitrary quantum-mechanical system unless one knows how to represent the corresponding Hamiltonian in the form $$ H = A^\...
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3answers
197 views

Can we derive Schrödinger equation from classical wave equation?

In classical mechanics wave equation is $$y=A\sin(kx-\omega t)$$ $y$=instantaneous displacement, $A$=maximum displacement, $\omega$=angular velocity, $x$=position of particle, $k$=wave number Now in ...
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2answers
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Meaning of the term $V(x)$ in the Schrodinger equation [closed]

I'm new to quantum mechanics and I am currently trying to understand finite potential well (although my question is not specific to finite potential well ). In the Schrodinger equation, many texts ...
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2answers
329 views

In a spherically symmetric central potential why do we look for eigenfunctions of the angular momentum operator?

In finding the solutions to the wave equation for a spherically symmetric potential $V(r)$, we look for the eigenfunctions of $\hat L^2$ and $\hat L_z$ operators. However, what is the reasoning behind ...
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1answer
50 views

Consistency of time-dependent and time-independent perturbation theory

I am confused as to how time-independent and time-dependent perturbation theories in QM give consistent results at the instant the perturbation is switched on. Suppose I have a two-level system which ...
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2answers
66 views

Schrodinger equation in momentum representation with position-dependent effective mass

I'm trying to convert Schrodinger equation with position-dependent effective mass (PDEM) to momentum representation, and I'm not sure how to apply the kinetic energy operator. In position ...
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Schrodinger equation for an atom [closed]

Does schrodinger equation describes only electrons and subatomic particles or also the atoms?
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66 views

$\hbar \approx 0$ and the spread of QM wave function

Is there a direct mathematical method to show that if a quantum wave funtion is initially sharply localized, then it will stay sharply localized if $\hbar \approx 0$? In that case the Ehrenfest ...
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1answer
53 views

Pure imaginary Schroedinger wave function

I know that the solutions to the time-dependent Schrodinger equation are always linear combinations of the form $$ \Psi(x,t)=\sum_n c_n e^{-iE_nt/\hbar} \psi_n(x) $$ If $ \Psi(x,0) $ is PURELY ...
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What are the experiments performed to determine the position of an electron inside an atom to verify the probability wave function data?

What are the experiments performed to determine the position of an electron inside the atom to verify the probability wave function data? Is it possible to do those experiments in real life?
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Time-independent Klein-Gordon PDE

Given the KG PDE: $$\psi_{tt} - \psi_{xx} + m^2 \psi = 0.$$ Wikipedia describes the time-independent variant of this as just setting $\psi_{tt}=0$. My question is this: For the Schrödinger ...
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2answers
106 views

Conceptual understanding of Schrödinger equation

So I followed this lecture: https://www.youtube.com/watch?v=qu-jyrwW6hw which starts of with the statement: If you have a Schrödinger equation for an energy eigenstate you have $$-\frac{\hbar}{2m}...
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1answer
124 views

Infinite square well: wall with infinitesimal thickness

Given an infinite square well, it doesn't matter how thick the wall is, the particle is trapped inside the two walls. If we make the wall of arbitrarily small but finite thickness, the particle is ...
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1answer
49 views

Scaling Problem with Variational Method

$\def\braket#1{\langle#1\rangle}$ I am attempting to solve a particular Hamiltonian by variational method. The wavefunction that I have selected is as follows: $$ \Psi = Ne^{\frac{-kr}{2}}\sum_{i=0}...
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1answer
62 views

Numerical approximation of the wavefunction in a delta-potential [closed]

I am trying to approximate the wavefunction of a particle in a delta potential $U(x) = -U_0 \delta(x)$ with $V_0 \gt 0$. I am using the following formula to calculate the wavefunction: $\psi(x+\Delta ...
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4answers
159 views

Does the Schrodinger wave function associated with a non-moving free particle change in time?

I'm a bit confused by an answer given on this question. In the answer with the animation of a moving free (chargeless) particle and a non-moving free particle (or a free particle with a non-zero ...
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1answer
35 views

Why do you need to add variables when multiplying through by a constant? [closed]

I was reading the following paper: http://vixra.org/abs/1206.0055 when I came across equation 9 on page two. The author claims that when multiplying equation 8 through by the constant A, you need to ...
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1answer
27 views

Double Gaussian well bound states parameters

I'm currently simulating a one-dimensional double Gaussian well potential numerically, and have been asked to find parameters corresponding to the overlap in potential and wave functions from my ...
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1answer
60 views

Help me make sense of the spectrum for the quantum wave function of an infinitely hard equilateral triangle

I'm trying to solve the spectrum for a equilateral Tetrahedron with infinitely hard walls. My first guess is to sum up a infinite amount of separable solutions to match the boundary conditions on the ...
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2answers
133 views

Non-Relativistic Limit of Klein-Gordon Probability Density

In the lecture notes accompanying an introductory course in relativistic quantum mechanics, the Klein-Gordon probability density and current are defined as: $$ \begin{eqnarray} P & = & \dfrac{...
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2answers
156 views

E.L. Equations in QFT

In QFT, we use the Lagrangian to construct the Hamiltonian, and in the Interaction Picture (with regards to the Free Field Hamiltonian) use the full Hamiltonian to calculate the changes in the field (...
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4answers
95 views

Does a particle with infinite energy escape an infinite well?

Currently, my modern physics class is going over particles in finite and infinite wells, general quantum formalism, and tunneling. What happens to a particle as it gains an infinite amount of energy? ...