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.

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81 views

Derivation of group velocity using Fourier transform

The aim is to determine the group velocity of a wave packet with the general form $$\Psi\left(x,t\right)=\frac{1}{\sqrt{2\pi}}\int_{-\infty}^{\infty} \phi\left(x\right)e^{i\left(kx-\omega ...
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25 views

Why can a free particle not exists in a stationary state? [duplicate]

In normalising the complete wavefunction of a free partcile; $V(x)=0$ we arrive at $\int_{-\infty}^{\infty}\Psi_{k} ^\dagger\Psi_{k}dx=|A|^{2}\int_{-\infty}^{\infty}dx=|A|^{2}\left(\infty\right)$ ...
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20 views

For a quantum free particle, would it be possible to relate the wavefunction $a_E(E)$ in energy basis and $a_p(p)$ in momentum basis?

The energy for a free particle has continuous energy eigenvalues $E$. Let $u(E,x)$ be its energy eigenstates in position basis. Its wavefunction $\psi(x,t)$ can be expressed as \begin{align} ...
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51 views

How can we fix the constant of the energy eigenstates of a quantum free particle such that they satisfy the orthonormality condition?

For a quantum free particle, the momentum and energy eigenstates are compatible. The constants of the momentum eigenstates are fixed by their orthonormality. Similarly, how can we fix the constant for ...
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84 views

Tricky particle in an infinite potential well question

For a particle in an infinite square-well potential in an energy eigenstate, the probability distribution relating to outcomes of position measurements vanishes outside the square well and takes a ...
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50 views

In quantum descriptions of atoms why are observables (which we derive from the wave function) attributed to electrons?

For example the orbital angular momentum, for the hydrogen atom. Is this the total angular momentum of the atom(electron and proton) or just the electron? I am asking because, I am learning about how ...
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3answers
80 views

Complex conjugate of hydrogen ground state wave function [closed]

For hydrogen atom ground state we know . I want to know the complex conjugate of .
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0answers
32 views

Find the eigen state of angular momentum for a trial wave function [closed]

If I have a simple trial wave function like this, $$\psi(r_x,r_y,r_z)=\exp(-v_1(r_x^2+r_y^2)-v_2r_z^2)$$ Can you tell me the detials for how to get the eigen state for $J=0$ for this wave function ...
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1answer
75 views

Physical interpretation of the constant coefficient appearing in solution to the Schrodinger equation

The product solution to the Schrodinger's equation is $$\Psi_{n} \left ( x,t \right )=\psi\left ( x \right )\phi\left ( t \right )$$ By superposition, the solution becomes $$\Psi \left ( x,t ...
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55 views

Quantum Backaction Question

Question about quantum back action in hypothetical scenario: We know that, at $t_0$, a certain kind of particle, with spin initially prepared to be “spin right” in the x basis, goes through a ...
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47 views

Propagation of momentum space wave function in linear potential [closed]

Under linear potential which denoted as $$V(x)=-Fx$$ Momentum space eigenket at time $t=0$ and some time $t$ has a relationship of $$\phi(p,t) = \mathrm{exp}\left[\frac{(p-Ft)^3-p^3}{6m\hbar ...
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52 views

Infinite square well physical interpretation

In quantum mechanics, the description of the infinite square well is given with the potential energy defined as $$V(x) = \begin{cases} 0 & \text{if } 0 \leq x \leq a,\\ \infty & ...
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1answer
95 views

Finite two-dimensional potential square well [closed]

I'm trying to solve the Schrodinger equation $$ -\frac{\hbar^2}{2m}\nabla^2 \psi+V(x,y)\psi(x,y)=E\psi(x,y) \tag{1}$$ for the finite two dimensional potential square well, that is, where ...
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0answers
31 views

Probability of measuring an observable $P$ in state $f$, computation [duplicate]

I have state vector $$f(x)=e^{-|x|+ix}$$ and observable $$P=-i\frac {d} {dx} $$ probability that measurement of $P$ in state $f$ will be in $[-1,1]$ I am stuck on this step. I dont know how to take ...
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1answer
42 views

Energy Expectation Value

I had an assignment question in which I was asked to calculate the expectation value of energy, $\langle E\rangle (t),$ and in the solution to it, the following was stated: \begin{align*} \langle ...
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2answers
59 views

Transmission and reflection amplitudes for delta potential Schrodinger equation

I hope this question is not too straightforward for this Q&A site. I have been reading a set of notes in which the transmission and reflection amplitudes for the delta potential Schrodinger ...
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1answer
41 views

Why is the wave function an element of the function space? [closed]

The general wave function is of the form $$\Psi \left ( x,y,z,t \right )=\psi \left ( x,y,z \right )T\left ( t \right )$$ Solving via separation of variables and finding the product solutions ...
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1answer
332 views

How is it possible to pull out derivatives of a wavefunction?

In an early derivation, the following equation was stated: $$\frac\partial{\partial t}\lvert\psi\rvert^2 = \frac{i\hbar}{2m}\biggl(\psi^*\frac{\partial^2\psi}{\partial x^2} - ...
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4answers
135 views

Is there a reason why probability density is written as $\psi^*\psi$ instead of $\psi\psi^*$?

As the title states, I see $|\psi|^2$ written as $\psi^*\psi$ instead of $\psi\psi^*$. Are both correct or is there a reason behind it? As far as I'm aware, the only time I see this sort of ordering ...
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1answer
177 views

Ground state of a particle in a ring - angular momentum is 0, but it is 'rotating' anyway?

Particle in 1D ring is a textbook problem, but there is one thing I don't understand - if the ground state is considered to have zero angular momentum, then its energy is also zero. And the ...
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86 views

Do wave functions really belong to $L^2$ space, or do we need to restrict our physical Hilbert space even further?

I am beginning to study quantum mechanics and I got stuck right at the beginning. I am trying to prove that the time derivative of the expected value of momentum of a particle is the (negative) ...
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1answer
84 views

How do we find the number of bounded states in this potential?

for the potential $$V(x)=-\frac{1}{1+\frac{x^2}{m^2}}$$ we can approximate the wave function and bounded state accurately for $x << m$ as simple harmonic oscillator, so what are we gonna do if ...
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87 views

Guess the wave function in a given potential

Are there any techniques in guessing the ground state wave function in any given potential? For example, for a given potential like $$ \frac{1}{1-x^2}$$ or $$ \frac{1}{1-x^3}~?$$ I know wave ...
2
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1answer
212 views

Analogy to Fourier transform in spherical coordinates with boundary at a certain radius

Suppose, we have a wavefuction $\phi(\vec{x})$ which is restricted in a sphere, with the spherical boundary condtion $$\phi(\vec{x}=R)=\phi_0.$$ How can I do the 'Fourier transformation' as the case ...
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2answers
85 views

Measurement of energy apparently violating the position-momentum Uncertainty Principle in a potential that does not depend on distance?

I am taking a beginning course in QM and I have learnt that the measurement of energy collapses the wavefunction of a particle to one of its energy eigenstates. But some misconceptions regarding this ...
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2answers
98 views

Normalisation of free particle wavefunction

The wavefunction $\Psi(x,t)$ for a free particle is given by $$\Psi(x,t) = A e^{i(kx-\frac{\hbar k}{2m}t)}$$ This wavefunction is non-normalisable. Does this mean that free particles do not exist in ...
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44 views

Additional quantum states of the infinite square well

The quantum states $\psi(x)$ of the infinite square well of width $a$ are given by $$\psi(x) = \sqrt{\frac{2}{a}}\sin\Big(\frac{n \pi x}{a}\Big),\ n= 1,2,3, \dots$$ Now, I understand $n \neq 0$, as ...
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2answers
90 views

Difference for boundary condition, particle in a box

When solving the simple problem of a free particle in a box of volume $V = L^3$, we can impose either periodic boundary conditions $\psi(0) = \psi(L)$ and $\psi '(0)= \psi'(L)$ either strict boundary ...
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3answers
99 views

Modern explanation of the Young experiment with Quantum Field Theory?

In the Young double slit experiment it is possible to detect the arrival of individual photons as well as an interference pattern. It doesn't makes much sense to me that something could be either a ...
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1answer
92 views

Bound states of Dirac Delta function in infinite well

If there is a potential of $-\alpha\delta(x)$ for $-a<x<a$ and $\infty$ elsewhere, and the energy of the system is less than 0, then I'm trying to find the wave function. From the Schrodinger ...
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0answers
32 views

How come an electron's wave function being nonzero at far distances doesn't mean it can travel faster than light? [duplicate]

I think the wave function of a free electron is nonzero almost everywhere. In particular there are regions of space arbitrarily far away where the electron has positive probability of being found. If ...
2
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1answer
78 views

What happens to the wave function of a particle immediately after measuring its energy?

For this question, I will be adhering to the Copenhagen interpretation (since that's what I've learned in university so far). For the sake of brevity/clarity, also, assume the Hamiltonian here has ...
2
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1answer
46 views

Antisymmetry requirement for the total wavefunction

My understanding is that if we are dealing with a system of two electrons, the total wavefunction needs to be antisymmetric only when the two electrons have same value of n and l ( i.e. they are ...
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2answers
51 views

Measurement of position after collapse of a wavefunction

Suppose I have a wavefunction which collapses to a certain eigenstate after a measurement of energy. In that state, I perform a calculation of position and obtain a certain position value, say $x_0$. ...
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43 views

Showing two wavefunctions are proportional to one another [duplicate]

I am struggling to answer the following question: Let ψ₁(x) and ψ₂(x) be normalisable energy eigenfunctions for a particle of mass m in one dimension moving in a potential V(x). Suppose that ψ₁ and ...
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106 views

Schrodinger equation violates mathematics?

By the Hamiltonian formalism of quantum mechanics, given a quantum system in a state $\Psi$ in a Hilbert space $\mathcal H$, the state will instantaneously evolve in time according to ...
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2answers
112 views

How do we decide whether an electron orbital has a non-zero or zero probability of lying inside the nucleus of an hydrogen atom?

How do we decide whether an electron orbital has a non-zero or zero probability of lying inside the nucleus of an hydrogen atom? It is mostly from the radial function, as to what I think but how ...
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34 views

Is the wavefunction of particles inside a gas spread or localized?

For an individual free particle that starts localized, the wave function packet spreads over time, so the particle becomes less localized. Suppose now that we have a gas of those particles inside a ...
3
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2answers
212 views

Why does the wavefunction have to be continuous in the presence of a Dirac delta potential?

Considering the time-independent Schrödinger equation, I can see for a finite potential, why the wavefunction has to be continuous, I can also see why the first derivative of the wavefunction is ...
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2answers
196 views

Time dependent and time independent Schrödinger equations

I'm trying to understand the relation between the time dependent and time dependent Schrödinger equations. In particular, we know that the TDSE is $$H\Psi=i\hbar \frac{\partial \Psi}{\partial t}$$ ...
2
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2answers
150 views

Minimum uncertainity

I'm confused in finding the condition for minimum uncertainty, The author in the book I refer goes on saying that $|g\rangle=c|f\rangle$ is the condition for minimum uncertainity for some constant ...
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52 views

solutions of wave equation with cubic term

Does the following equation $$ \nabla^\mu \nabla_\mu \psi + a \psi^3 = b \psi $$ where $\psi$ is a real function, $a$ and $b$ are real constants, have other solutions that extend beyond a one ...
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1answer
93 views

Infinite potential well question with wave function $\psi (x) = (x -a/2)^2$

In an infinite potential well with width $a$, a particle in this potential well is at state with wave function is $\psi (x) = (x -a/2)^2$ (not normalized). If you measure the energy of ...
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1answer
78 views

Confused on how to interpret the energy eigenfunction of Hydrogen

So here is an image of the third lowest energy eigenfunction of an electron in a hydrogen atom: Image from http://imgur.com/Lu4MocL I understand well the eigenfunctions given by Schrodinger's ...
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1answer
43 views

Spinor expectation value and measurement

I have a question about the difference between expectation value and probability of measurement. consider the spinor $\zeta = [-3\ \ 4i\ ]^T$ . The expectation value of $S_x$ is zero because : ...
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103 views

Coefficients and wavefunction in quantum mechanics

In general quantum mechanics we represent the state of a system with a state vector $| \psi \rangle $ in some Hilbert space in some base. Assuming a complete discrete set of bases vectors $ |n \rangle ...
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3answers
91 views

Quick way to compute $\langle n^{'}l^{'}m^{'}|r^k|nlm \rangle$, $k \in I$; $|nlm\rangle$ is $H$ atom eigenfunction [closed]

I want to compute quickly (using maybe some scaling arguments) $\langle n^{'}l^{'}m^{'}|r^k|nlm\rangle$, where $k \in I$. $|nlm \rangle$ is the eigenfunction of the Hydrogen atom ($H$). Example: ...
0
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3answers
181 views

How can quantum wavefunctions be smooth/continuous when particles are created/destroyed/changed?

My (admittedly limited) understanding of the Schrodinger equation tells me that the vector differential operators are only meaningful over a differentiable phase space. For example, if the dimensions ...
2
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1answer
119 views

Is a non-degenerate wavefunction real or complex?

In this video it is stated that: It can easily be verified that the wavefunction of a non-degenerate quantum mechanical system will be real. However the presenter does not explain why this ...
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71 views

Why must the probability be the integrated square modulus of the wave function [duplicate]

Quantum mechanics uses the wave function to calculate probabilities by taking the square modulus of the wave function as requirement by Max Born. Why should this (squaring of the wave function) be so, ...