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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19 views

Normalization of Airy function [on hold]

I hv to normalize this wave functiom but due to airy function,I hv no idea how to normalize this.I also check the landau and lifshitz.vol.3.Quantum mechanics non relativistic theory,pg#74, I didn't ...
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55 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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32 views

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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56 views

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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21 views

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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1answer
26 views

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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1answer
110 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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103 views

A Schrödinger Love Poem? [on hold]

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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51 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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41 views

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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55 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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19 views

Transmission coefficient in step potential [closed]

I am calculating the transmission coefficient from a scattering wave coming from $-\infty$ into a step potential given by: $$ V(x) = \left\{ \begin{array}{cc} 0 & x\leq 0 \\ V_0 &...
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1answer
83 views

Energy eigenvalue with Potential $-e^2/ x$

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 ...
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213 views

Normalization constant of a planar wave

As we know for the plane waves ( $ae^{i k x}+b e^{-i k x}$), the normalization constant can be easily obtained from the integral $\int^{x_{2}}_{x_{1}}\psi^{*}\psi dx=1$ by the relation $|a|^{2}+|b|^{2}...
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588 views

Is the evolution operator well-defined mathematically?

We know that in order to solve the time-dependent Schrodinger equation $i\partial_t \psi = H(t) \psi$, we need the evolution operator $$U(t) = T \exp{\left(-i\int_0^t H(t')dt'\right)}$$ where $T$ is ...
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31 views

Potential barrier, vertically shifted

I'm attempting to solve the 1D Schrodinger Equation, approaching a potential barrier defined as follows: $$V(x) = \begin{cases}-V_0&\quad\text{for}\quad x<0 \\0&\quad\text{for}\quad x>0\...
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92 views

Can one derive Schrodinger's Equation from quantum information theory?

I know that some people think that quantum information theory/science is fundamental physics. I also know that there are many definitions, theorems and rules in the field of quantum information. They ...
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31 views

Bound states and parity for a arbitrary potentials

If we are given an arbitrary potential, and we are asked to find bound states and parity, what would be usual strategy to do that? Let's we have a potential given: $$-\frac{A}{y^2+a^2} -\frac{A}{(y-...
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1answer
59 views

How to determine initial quantum state? [closed]

A particle in an infinite square well has its initial wave function an even mixture of the first two stationary states: $$\psi(x,0)=A(\psi_1(x)+\psi_2(x)) $$ As you may know, for $\psi(x,t)$ we ...
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54 views

Issue with solving for the wavefunction of a simple infinite potential

For the potential given by $V(x)=\left\{\begin{array}{ll}{\infty} & {x<0} \\ {-V_{0}} & {0<x<a} \\ {0} & {a<x}\end{array}\right.$ I am trying to solve for the wavefunction. ...
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1answer
90 views

How can we prove that the Hamiltonian for any quantum system is Hermitian? [closed]

By applying partial time derivative to $$\psi_t \rightarrow U \psi_{t_0}$$ we end up with an expression for the Hamiltonian $$H = i\hbar\frac{\partial U_{t}}{\partial t}U^{\dagger}_{t}$$ where $U$ is ...
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In the Schrödinger equation, can I have a Hamiltonian without a kinetic term?

To find out the stationary states of Hamiltonian, we will be finding the eigenvalues and eigenstates. Is there any condition that form of the Hamiltonian should be like, $$\hat{H}=\hat{T}(\hat{p})+\...
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3answers
56 views

Expected value of Momentum in a square infinite well [closed]

Say I have a particle with mass $m$, in a potential infinite well centered at $x=0$ with length $d$ which wave function at $t= 0$ is represented by: $$\Psi(x)=\begin{cases} \frac{1}{\sqrt{2}}\left[\...
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What is the easiest system to take the matrix representation of a Hamiltonian?

To understand how the unitary operator, preserve the inner products, I wanted to explore the unitary operator as a matrix. Now the equation for the unitary operator (time evolution operator) has a ...
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91 views

What potentials have infinitely many bound states? [on hold]

Some potentials have only finitely many bound states (the finite square and delta function are two good examples) Others have infinitely many bound states (for example the infinite square well and $1/...
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2answers
69 views

Why doesn't the amplitude of a wave-function fall off to zero immediately at a potential barrier?

When a wave function in QM potential well problems interact with a potential barrier with height more than the energy of the wave, the amplitude of the wave doesn't immediately falls off to zero, ...
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3answers
77 views

Negative energy in bound states of a particle in a finite potential well

Consider you have a particle in a finite potential well as depicted in the photo attached. Now we have three regions: $$V(x) = \begin{cases} 0, & \text{for } x<-a & (1)\\ -V_0, & \...
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1answer
33 views

Expressing unitary operator of time as a matrix

Can someone please show the full calculation if we were to experess as a unitary matrix
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15 views

Factorizing spin part and space of many electron wave function of an atom's ground state

I am trying to write the ground state wave function of a 10 electron atom as a product of space part and anti-symmetric spin part. $$1s\uparrow,1s\downarrow$$ $$2s\uparrow, 2s\downarrow$$ $$2p_{x}\...
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1answer
110 views

Dyson Series Iteration - Gives Exact Solution?

When we derive the Dyson series for usage as the time evolution operator in the case of a time dependent Hamiltonian, we start with the equation: \begin{align}\hat{U}_I(t,t_i) = 1 - \frac{i}{\hbar}\...
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39 views

Schrodinger equation: If $V(x)=V(-x)$ then prove that $\psi(x)=\psi(-x) $ or $\psi(x)=-\psi(-x)$ [duplicate]

The title explains itself. If the potential is an even function then prove that wave function is either odd or even. I set $-x$ in Schrodinger equation and find out that $\psi(-x)$ is also a solution ...
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35 views

Stationary State of Quantum Mechanics [duplicate]

Why for every normalized solution to the time-independent Schrödinger equation $E$ must exceed the minimum value of $V(x)$?
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1answer
67 views

Understanding the quantum mechanical state vector

According to Griffiths, there is a general state vector $|s(t)\rangle$ that encodes the state of the system. He also says that we take $\Psi(x, \ t) \ = \ \langle x | s(t) \rangle$. Would then mean ...
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76 views

States for derivatives of wave function?

Given a wave function $\psi_t(x)$. The quantum state of a system at time t can be written as the sum of basis states multiplied by the amplitude: $$|t\rangle = \int \psi_t(x)|x\rangle dx^3$$ What ...
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Proving the preservation for the norm of a wavefunction

How can we prove the preservation for the norm of the wave-function for a specific hamiltonian (say a spin 1/2 particle) for all times?
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30 views

Probability density of time-dependent wave functions

Why is it so that probability density of eigenfunctions of time-dependent schrodinger equation are time independent while that of general wave functions (which are a combination of the eigenfunctions) ...
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75 views

Is resonance an energy eigenstate?

In the particle physics book of Martin & Shaw, they used QM to derive the decay distribution, namely, breit wigner formula. What confused me was that, here they assumed the resonance state $\psi_0 ...
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1answer
54 views

Can the spectra of helium atom be solved by Schrodinger equation?

Can the Schrodinger equation work out the spectra of helium atom, not all, but part of them?
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69 views

Why does the wave function of a non relativistic particle flatten out over time?

The Hamiltonian I used is the classical one with no potential energy: H=p^2/2m $$i \hbar \frac{\partial \psi}{\partial t} = -\frac{\hbar^2}{2m} \frac{\partial^2 \psi}{\partial x^2} $$ I want to gain ...
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1answer
40 views

Conversion of the nonlinear schrodinger equation from $\partial_zE$ to $\partial_tE$

While reading some papers about the nonlinear schrodinger equation (NLS) I noticed that the authors sometimes use (for the linear case) $$\partial_zE=\frac{i}{2k_0}\nabla^2E$$ and sometimes $$\...
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2answers
133 views

Does the Schrodinger Equation care about spin?

I have taken the non-relativistic limit of the Klein-Gordon and Dirac equation, and both have brought me to the Schrodinger equation. The Klein-Gordon equation describes spin 0 particles, and the ...
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1answer
26 views

Partition Function when the Energy States are Both Discrete and Continuous

Normally for statistical mechanics (in this example I will be only refering to the canonical formalism to keep things simple) we generally have a system that we solved the equations of motion for and ...
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1answer
46 views

Implementing the Hartree-Fock method in two dimensions from scratch

I am interesting in writing a complete code for the Hartree-Fock method to improve my understanding of it. It seems relatively complex in three dimensions, so I am wondering if it would be simpler in ...
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45 views

Independents fields and the Lagrange Density of Schrodinger equation [duplicate]

I have a doubt about the lagrangian of the Schrodinger equation. If we consider the wave function $\psi(\textbf{x},t)$ that satisfy the Schrodinger equation as a field, one way of construct the ...
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2answers
53 views

Solving time-independent Schrodinger equation with $V=0$

I recently came across this section in "Physical chemistry" by Peter Atkins and Julia Paula. There they discuss the solution of Schrodinger equation for a particle of mass m moving through a single ...
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Why is the Schrödinger field an annihilation operator?

The relativistic scalar field operator is not a ladder operator. Its commutation relations are $$\begin{align} \left[\hat{\phi}\left(\vec{x}\right), \hat{\phi}\left(\vec{y}\right)\right] = \...
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4answers
152 views

Can the Schrodinger Equation be Relativistic in it's General Case?

So as we all know the Schrodinger equation is not relativistic when written (constantless) as$$i\partial_t\psi=-\partial_{xx}\psi+V\psi$$due to it not being Lorentz invariant. However, if we consider ...
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Necessary and suficient conditions for Scattering to be Elastic

Pretty straightforward: what basic assumptions must we make in constructing a Scattering Theory, in Quantum Mechanics, in order for it to conserve energy of the incident particle (i.e. to be elastic)? ...
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34 views

Interaction picture counter rotating terms

In the interaction picture, we often do the rotating wave approximation where terms like $e^{i(\omega_1 + \omega_2)t}$ are ignored because they represent rapidly oscillating terms which ends up ...
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3answers
76 views

What is $x$ in the Time-Independent Schrödinger Equation? [closed]

I couldn't find this answered anywhere, even though it's absolutely central. What does the value $x$ stand for in the time-independent Schrödinger equation? It's the thing we solve for, but it doesn't ...