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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13 views

Deriving Rabi oscillations using the Heisenberg picture of QM

The semiclassical treatment of an simple two level atom in a resonant electromagnetic field is usually done in the Schrodinger/Interaction picture of QM, by assuming that the wavefunction of the atom ...
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1answer
1k views

Is the existence of a sole particle in an hypothetical infinite empty space explicitly forbidden by QM?

Suppose the universe is completely empty with one sole particle trapped in it. To simplify, I will only be looking at the one dimensional case. However, all arguments are applicable for three ...
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1answer
53 views

Can we have $E=0$ in Schrödinger's Equation?

I've read a little bit about zero-energy states, but I just don't get it. I'm just starting to study quantum mechanics and, at least for all the potentials I've seen until now (the most popular ones, ...
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24 views

Variational Method for one dimensional infinite square well [on hold]

I have an one dimensional infinite square well of width $a=1$, and I want to use variational method to produce the ground state energy. My trial wavefunction is ...
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0answers
16 views

Are there any specific examples of the application of Lewis-Riesenfeld procedure to time dependent Hamiltonians in QM?

Lewis-Riesenfeld invariant theory is a theory applicable to solve time-dependent Schrodinger equations. I have always encountered the theory related to the procedure, however never encountered any ...
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2answers
59 views

Use Runge Kutta method to solve schrodinger equation

The schrodinger equation in spherical coordinates after seperation of variables as a solution of hydrogen atom is given by $$ \frac{-\hbar^{2}}{2 \ m} \left[ \frac{1}{r^{2}} \frac{d}{dr} \left(r^{2} ...
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0answers
32 views

Solving Schrodinger's Equation in Bunimovich Stadium Boundary Condition [closed]

I need to solve, as mentioned, Schrodinger's equation in a Bunimovich stadium-shaped infinite potential well with Dirichlet BC Numerically (this isn't possible analytically). In order to do so, I need ...
3
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1answer
65 views

Limits used to find non-rel limit of the Klein-Gordon equation

I just have a question regarding assessing the non-relativistic limit of the Klein-Gordon equation. In the book I'm following (Quantum Mechanics by Bransden & Joachain) they use the limits (Chpt. ...
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37 views

Particle in a Spherically Symmetric Box [closed]

Problem The spherically symmetric potential energy of a particle with mass $m$ is given by $V(r)=0 $ if $ a<r<b$ $V(r)=\infty$ elsewhere where $r^2 = x^2+y^2+z^2$. (a) ...
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0answers
24 views

Matrix representation of the radial Laplace operator isn't symmetric (or hermition as a result)

I'm working with the cylindrical coordinates. I'm using the central difference to convert the radial part of Laplace operator into a matrix. $\nabla^2 u = \frac{\partial ^2u}{\partial ...
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2answers
54 views

Confusion of Schrödinger equation and complex conjugates

I have a similar question that was asked in the following link: (Schrödinger's Equation and its complex conjugate). But I find both the question and answers not specific enough. So let me ...
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1answer
19 views

Relation between probability density and transmission probability of a wavefunction?

Problems I did on current densities in elementary quantum mecanics course gives the answer contains transmission coeffecients, I am wondering is there any relation among them.
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76 views

Solving the Schrodinger Equation

What is meant by solving the stationary schrodinger equation? I have a well of width $L$ where $v(x)=-v$ inside the well and 0 outside I am trying to solve the schrodinger equation but i dont ...
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1answer
39 views

1D Scattering Phase Shift (Finite Well) - Unphysical?

I am calculating the phase shift from a 1-dimensional potential well. This seems extremely simple, but I am just getting so confused by it. Let there be a potential well of depth $V_0$ and spatial ...
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0answers
40 views

Galilean transformation of Schrodinger equation and momentum operator

Let $$ \left.\begin{aligned} t'&=t\\x'&=x-vt \end{aligned}\right\} \quad \Longrightarrow\quad \dot{x}'=\dot{x}-v $$ and therefore $p'=p-mv$. If $p'=-i\hbar\nabla' $, then ...
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0answers
32 views

Are $\psi ^{*}(x,t)$ and $\psi(x,-t)$ solutions of the same Schroedinger equation?

I have this question: Let $\psi(x,t)$ solution of the Schroedinger equation for a particle under a potential V(x) independent of time. Are $\psi ^{*}(x,t)$ and $\psi(x,-t)$ solutions of the same ...
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3answers
440 views

Interpretation of boundary conditions in time-independent Schrödinger equation

The time-independent Schrödinger equation: $$\ -\frac{\hbar^2}{2m}\frac{d^2\psi}{dx^2} + V\psi = E\psi$$ is second order, so we should expect the solution to have two "degrees of freedom" which can ...
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98 views

Why can we set the coefficient $c_- = 0$ in the solution of the quantum particle on a ring?

In the quantum particle in a ring problem, the general solution for the wavefunction, with $k = R \sqrt{2 m E / \hbar^2}$, $R$ being the ring radius, $c_{+, -}$ being constants, $E$ the energy, and ...
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1answer
44 views

Allowed Wave Functions of System

Given a single-particle system with Hamiltonian $H$, what constraints can be put on the wave function at a particular point in time $\psi(x)$? Of course $\psi(x)$ must obey boundary conditions given ...
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0answers
43 views

How to find the minimum value of potential in QM?

In MIT problem sets I followed a solution of an exercise which focuses on odd-parity energy eigenstates in finite square well. The point of problem is how to know or find the minimal value of ...
3
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2answers
141 views

Schrödinger equation in momentum space

In literature on an introduction to quantum mechanics which I am working through, there is a section which explains that a vector has different representations based on the basis you choose. It then ...
3
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4answers
93 views

What is $V(x)$ in Schrödinger's equation?

In the time-independent Schrödinger equation it is stated that $$-\frac{\hbar^2}{2m}\frac{d^2\psi(x)}{dx^2}+V(x)\psi(x)=E\psi(x)$$ And it is common to give $V(x)$ some standard "forms": the infinite ...
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0answers
24 views

Showing the transmission coefficient is valid

In a semiconductor device, electrons accelerated through a potential difference of 7V attempts to tunnel through a barrier of width 0.5nm and height 10V. Assume the potential is zero outside the ...
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1answer
52 views

Can there be a wave function that is physically possible but is non differerentiable (maybe even non-continous)?

The definition of a wave function demands continuity and differentiability so that it can satisfy the Schrödinger Equation. My question is whether this assumption is necessary for reality. Does ...
3
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1answer
59 views

Instantaneous energy eigenstates for forced harmonic oscillator

I'm interested in applying the adiabatic theorem to the forced harmonic oscillator with time dependent hamiltonian of the form: $$H(t) = \hbar \omega(a^{\dagger}a + \frac{1}{2}) - f(t)a - ...
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24 views

Choosing the boundary conditions knowing the potential

So I have to apply Numerov's algorithm to solve the Schrödinger equation in spherical coordinates using the potential $V(r)=-A e^{-r/a}$, where $A=32.7MeV$ and $a=2.18fm$. I have already ...
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1answer
40 views

Correlating two definitions of bound states in quantum mechanics

In Griffiths, he defines a bound state to be that stationary state for which the total energy E is such that $E<V(\pm\infty)$. Let $\psi(x)$ is a stationary state satisfying $E<V(\pm\infty)$ and ...
6
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1answer
138 views

Reflectionless potentials in quantum mechanics

Scattering on potential $$V(x) = -\frac{(\hbar a)^2}{m}\text{sech}^2(ax)$$ with 1D equation of Schrodinger is famous problem. It is dealt with in Problem 2.48 of Griffiths book or online here. It is ...
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1answer
64 views

Schrödinger equation in spherical coordinates using Numerov's method?

So I need to solve the Schrödinger equation using Numerov's method. However, the problem I have assigned to me is in spherical coordinates. As it seems, I only need to take into account the dependance ...
2
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2answers
149 views

Complex conjugate of the Schrödinger equation?

This might be a very simple question but I don't understand how to compute the complex conjugate of the Schrödinger equation: $$ i\partial_t \psi = H\psi $$ where $H$ is an hermitian operator. How to ...
2
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1answer
73 views

Is there a way to prove that a bound state wavefunction can always be chosen real for an arbitrary potential in Quantum Mechanics?

As we can prove many things that always (at least in introductory quantum mechanical problems) apply using an arbitrary potential (like that $E>V_{\rm min}$ or else the solutions are ...
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1answer
22 views

Gauge transformation of wave function of a system of stationary charges

Let's say we have a system of $n$ stationary charges interacting via Coulomb potential. Let's ignore possible external electromagnetic fields. Moreover the system is quantum, and its wave function is ...
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1answer
30 views

elelctric potential from a laser

The average intensity from elecreomagnetic waves are given by: $$I=c \frac{E_{av}^2 \epsilon_0}{2}$$ I want to find what strength of a laser one needs to apply to get below the binding energy of a ...
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1answer
32 views

Representing an ISW wavefunction graphically [closed]

I'm trying to decode this diagram given to us for an assignment. The description of the diagram is 'Consider a particle of mass m confined to a 1-dimensional square well, given graphically by the ...
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1answer
67 views

Deriving Schrodinger equation from QFT with the definition $\psi(\textbf{x},t)\equiv \langle 0|\phi_0(\textbf{x},t)|\psi\rangle$

In the book "Quantum Field theory and the Standard Model" by Matthew Schwartz, he uses the equation $$\partial_t^2\phi_0=(\nabla^2-m^2)\phi_0$$ (i.e., the Klein-Gordon equation for the free ...
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1answer
43 views

Particle in infinite square well with $\langle E \rangle = (1/4) E_1 + (3/4) E_4$

Suppose we have a particle in an infinite square well with $\langle E \rangle = (1/4) E_1 + (3/4) E_4$. I know that I can calculate the uncertainty in the particle's position by $\sqrt{\langle E^2 ...
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1answer
79 views

Calculating the energy of a particle using the Time Independent Schrodinger Equation [closed]

If we have a wave function $\Psi(x,t=0)$ which is a solution to the TISE for a zero potential in an infinite square well, would calculating the energy at $t = 0$ at a position be as easy as ...
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1answer
128 views

Griffith's Proof that a wave function will stay normalized is incorrect?

In Griffith's book, Introduction to Quantum Mechanics, in the equation: $$ \frac{\partial}{\partial t} \left| \Psi\right|^2 = \frac{i\hbar}{2m} \left( \Psi^* \frac{\partial^2 \Psi}{\partial x^2} - ...
3
votes
1answer
117 views

Significance of $i$ in the Schrödinger equation [duplicate]

There's an imaginary $i$ in the Schrödinger equation, which I guess is to define the position of the particle in a space-time involving a complex function. But what is the real physical significance ...
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2answers
75 views

Should there be a QFT evolution equation which has different orders of time and space derivatives?

I have seen many QFT equations like Klein-Gordon, Dirac, Weyl, Proca etc. But they order of the time derivative is always the same as the order of the space derivative. Shouldn't there be an equation ...
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0answers
25 views

How to estimate the ground state of a potential well when a confinement dimension is added

I have a finite harmonic potential where I trap an electron. The confinement length changes in size. Now, I'm interested in the ground state energy, so I have this 1D Poisson solver which gives me the ...
2
votes
1answer
152 views

Why is the propagator the Green's function for Schrodinger equation? [duplicate]

Sakurai says that the propagator is simply the Green's function for the time-dependent wave equation satisfying $$\left [ -\frac{\hbar^2}{2m} \triangledown ''^2+V(\mathbf{x''})-ih\frac{\partial ...
1
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1answer
30 views

Normalization of states of continuos spectra with complicated boundary conditions

Let's consider the following Schrödinger equation: $$\psi''(x)+k^2\psi(x)=0$$ with the following boundary condition $$\psi(0)+a\psi'(0)=0$$ $k$ is supposed to be larger that $0$. This equation is ...
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0answers
47 views

Wave function for step potential

Given the step potential $$V(x)=\begin{cases} 0~~~~~~~~\text{if }~~x \leq 0 \\ V_0~~~~~~\text{if }~~x > 0 \end{cases}$$ Consider the case where $E < V_0$. In this region $x \leq 0$ we have ...
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4answers
211 views

Connection between Hamiltonian version of the least action principle and probability amplitude in the Schrödinger equation

If I'm not mistaken, Schrödinger was influenced to look at wave equations because of de Broglie's assertion about particles having a wavelength. He started with the Hamiltonian equation which is ...
2
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2answers
82 views

Normalization of a wave function in quantum mechanics

A more simple question, so I am watching a quantum mechanics lecture on potentials of free particles and am doing the general solution of schrodinger's stationary equation for a free particle when I ...
2
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1answer
53 views

Why is the reflection coefficient in quantum mechanical scattering defined this way?

In Griffiths' "Introduction to Quantum Mechanics, second edition" section 2.5.2, p. 73, he states: For the delta-function potential, when considering the scattered states (with $E > 0$), we have ...
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1answer
27 views

How to tell when exponentials are real valued? (Barrier Potential)

Where $k_1=\frac{\sqrt{2mE}}{\hbar}$ and $\alpha=\frac{\sqrt{2m(V_0-E)}}{\hbar}$ I'm quite confused as to why the exponentials in regions I and III are complex functions while in region II the ...
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0answers
27 views

Magnus Expansion in Floquet theory

I wonder how to obtain the second equality as follows in Eq. (44) of http://www.tandfonline.com/doi/abs/10.1080/00018732.2015.1055918?journalCode=tadp20 \begin{eqnarray} ...