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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Should the differential of a wavefunction have a time partial derivative?

In chapter 1 of Griffths' QM text, he shows that $\frac{\mathrm{d}}{\mathrm{d}t}\int_{-\infty}^{\infty}|\Psi|^2\,\mathrm{d}x=0$ by noting $$\begin{align} \frac{\mathrm{d}}{\mathrm{d}t}\int_{-\infty}^...
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No well-defined frequency for a wave packet?

There are similar questions to mine on this site, but not quite what I am asking (I think). The de Broglie relations for energy and momentum $$ \lambda = \frac{h}{p}, \\ \nu = E/h .$$ equate a ...
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125 views

Help needed to understand “On the reality of the quantum state”

I am having trouble to understand the reasoning in the following paper, On the reality of the quantum state. MF Pusey, J Barret and T Rudolph. Nature Phys. 8, 475–478 (2012); arXiv:1111.3328. ...
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Has anyone published the procedure to generalize ladder operators for any potential in Schrodinger's equation?

I know that the ladder operator for the quantum harmonic oscillator \begin{align} H\psi_m = \left(\dfrac{p^2}{2m}+\dfrac{1}{2}m\omega^2x^2\right)\psi_m=E_m\psi_m \end{align} is \begin{align} A = \sqrt{...
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Orthonormality of Radial Wave Function

Is the radial component $R_{n\ell}$ of the hydrogen wavefunction orthonormal? Doing out one of the integrals, I find that $$\int_0^{\infty} R_{10}R_{21}~r^2dr ~\neq~0$$ However, the link below says ...
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Effect of pressure increase on electron orbital wave functions

One of my nuclear physics exercises was to find out if increasing the pressure of a sample of $^{7}\textrm{Be}$ would increase the chance of electron capture to $^{7}\textrm{Li}$ occur. My reasoning ...
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446 views

Is the electron wave function defined during photon emission

I have heard the term quantum leap to describe the (instantaneous?) transition from a higher energy orbital to a lower energy orbital. Yet, I understand that this transition time has now been ...
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How do we know that electron wave function extends to infinity?

Why do physicists assume this? Is it a proven fact that wave function extends to infinity or just a theory? Would it make sense if they didn't extend to infinity?
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846 views

Degeneracy in one dimension

I'm reading this wikipedia article and I'm trying to understand the proof under "Degeneracy in One Dimension". Here's what it says: Considering a one-dimensional quantum system in a potential $V(...
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Boundary conditions from single-valuedness of spherical wavefunctions

This question is a follow-up to David Bar Moshe's answer to my earlier question on the Aharonov-Bohm effect and flux-quantization. What I forgot was that it is not the wavefunction that must be ...
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Relativistic contraction for a wave packet and uncertainty on momentum

Consider an electron described by a wave packet of extension $\Delta x$ for experimentalist A in the lab. Now assume experimentalist B is flying at a very high speed with regard to A and observes the ...
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Do bras and kets have dimensions?

I'm trying to understand more intuitively what bras and kets are, but some aspects of them remain a mystery to me. We usually think of $\psi (x)$ as having dimension of $[1/\sqrt{L}]$ so that ...
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Is the ground state closest to the uncertainty relation? [duplicate]

For simplicity, suppose we are only talking about discrete energy levels, ie, bound state case. The energy levels are $E_1, E_2\cdots$, and the corresponding wave functions are $\psi_1, \psi_2 \cdots$....
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Is there a direct physical interpretation for the complex wavefunction?

The Schrodinger equation in non-relativistic quantum mechanics yields the time-evolution of the so-called wavefunction corresponding to the system concerned under the action of the associated ...
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When Eigenfunctions/Wavefunctions are real?

When the Hamiltonian is Hermitian(i,e. beyond the effective mass approximation), generally under which conditions the eigenfunctions/wavefunctions are real? What happens in 1D case like the finite ...
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756 views

Can the chance of finding a particle diminish over time?

Let's assume we have a wave function described by a wave equation and it is a function of space and time $\psi : \mathbb{R}^4 \rightarrow \mathbb{C}$. This function needs to be normalized, so if I ...
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562 views

Schroedinger equation for hydrogen atom

I have got a problem understanding the meaning of the Laplace operator in the Schrödinger equation for the hydrogen atom. $$\Big(-\frac{\hbar^2}{2m_e} \Delta_{r_e} - \frac{\hbar^2}{2M_P} \Delta_{r_p} ...
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358 views

Where to place the operator?

I believe it's standard to place the operator in between the conjugate of the wavefunction and the wavefunction itself. For instance, $$\langle p\rangle = \int_{-\infty}^{\infty}\Psi * \frac{\hbar}{i}...
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Same quantum states represented in different basis

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 and then ...
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Where the time-dependent wavefunction $\Psi(\vec{x},t)$ lies?

Supose $\vec{x}=(x,y,z)\in \mathbb{R}^3$. The state of a physical system is described by the function $\Psi(\vec{x},t)$, where it must satisfy $$\int_{\mathbb{R}^3} d^3\vec{x}\;\vert\Psi(\vec{x},t)\...
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Is the free electron wavefunction stable?

The wavefunction of a free electrons is variously described as a plane wave or a wave packet. I am fairly happy with the wave packet, as it is localised. But if we change to the electron's rest ...
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State of a system in Quantum Mechanics and state vectors

I'm taking a course in Quantum Mechanics and there is something I'm not being able to fully understand. On more elementary courses on Quantum Mechanics I've been told that the idea of Quantum ...
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Can the expectation value of the square of momentum be negative?

I've been solving a problem in quantum mechanics, and I was deriving the standard deviation of $P$, knowing that $\langle P\rangle=0$. Because $\Delta P=\sqrt{\langle P^2 \rangle - \langle P \rangle ^...
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Does $\lvert\langle p\lvert\psi\rangle\rvert^2$ have any meaning at all?

I used to think $\lvert\langle p\lvert\psi\rangle\rvert^2$ had the meaning of some likelihood of the particle's momentum being $p$ (within some tolerance interval $\Delta p$). Now I'm just confused. ...
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How does a Wavefunction collapse?

I have been wondering and researching... How does a wavefunction collapse into one state?More specifically, what conditions cause a wavefunction for a quantum particle to collapse? Does this have to ...
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Help me understand the first equation in Landau & Lifshitz's Quantum Mechanics

While I've covered a basic course in Quantum Mechanics, I'm self-studying Landau & Lifshitz's book to help me understand what's going on. Unfortunately, I'm stuck on the very first equation in ...
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Born Interpretation of Wave Function

I have just started Griffiths Intro to QM. I was studying Born's interpretation of Wave function and it says that the square of the modulus of the wave function is a measure of the probability of ...
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3D Quantum harmonic oscillator

For an isotropic 3D QHO in a potential $$V(x,y,z)={1\over 2}m\omega^2(x^2+y^2+z^2).$$ I can see by independence of the potential in the $x,y,z$ coordinates that the solution to the Schrodinger ...
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Isn't the 'slit' in a double-slit experiment also a wave?

I'm new to QM so excuse my naivety. I was watching an online MIT QM course that described the double-slit experiment (with electrons) when it occurred to me that I have a question. In the video, the ...
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Linear vs. quadratic dispersion relation

In wave mechanics the dispersion relation between frequency $\omega$ and wave number $k$ is linear: $$\omega_n=c k_n$$ But in quantum mechanics, based on Schrödinger's equation, one can show that we ...
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Why can we leave off half of the general solution?

In these pdf notes, it says at the bottom of the first page and beginning of the second: [...] whose solution is: $$\Psi(\theta) = c_1 e^{i\omega\theta} + c_2 e^{-i\omega\theta}$$ Since we are ...
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What is the physical reason behind linearity of Schrodinger's equation?

What is the physical reason for Schrodinger equation to be linear? Though in physics many interactions or dynamics are found non linear.
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Can a wave possess spin?

Since a matter wave is associated with a particle in quantum mechanics, does the wave spins? I mean, can we visualize the spinning of wave or is it possible that the wave spins?
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Comparison of 1D and 3D wave functions

When discussing the Schroedinger equation in spherical coordinates, it is standard practice in QM handbooks to point out that the radial part of the 3-dimensional wave equation bears a strong analogy ...
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A quantum particle which is almost at rest but whose position is random!

Assume a particle is given by a quantum state which is constructed in such a way that it is equally probable to find it anywhere in an fixed interval $(0,L)$ but has arbitrarily low velocity. The ...
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Born Oppenheimer Approximation: Why can any molecular state be represented as a linear combination of electronic states?

in the Born Oppenheimer Approximation, one expands the molecular wavefunction $\Psi(x,X)$ in terms of the electronic wavefunctions $\phi(x;X)$: $$\Psi(x,X)= \sum_k(c(X)_k\phi(x;X)_k)$$ ($x$ are the ...
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Why is wave function so important?

I am almost sure that the wave function is the most important figures in modern physics book. On the other hand I know that wave function even do not have a physical meaning it self alone! Why is ...
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Where does the wave function of the universe live? Please describe its home

Where does the wave function of the universe live? Please describe its home. I think this is the Hilbert space of the universe. (Greater or lesser, depending on which church you belong to.) Or maybe ...
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Dirac Equation in RQM (as opposed to QFT) is written in which representation?

In introductory Quantum Mechanics treatments it is common to see the Schrödinger's equation being written, simply as: $$-\dfrac{\hbar^2}{2m}\nabla^2\Psi(\mathbf{r},t)+V(\mathbf{r})\Psi(\mathbf{r},t)=...
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Is $\phi_n =\left\langle \vec r | n \right\rangle $ the photon wave function?

I am a bit confused about this issue and I am still not clear whether is there is a photon wave function or not. Since we use Fock states $| n \rangle$ to represent the state of a quantized ...
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Why does $\ell=0$ correspond to spherically symmetric solutions for the spherical harmonics?

In quantum mechanics why do states with $\ell=0$ in the Hydrogen atom correspond to spherically symmetric spherical harmonics?
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Historical background of wave function collapse

I wonder what were the main experiments that led people to develop the concept of wave function collapse? (I think I am correct in including the Born Rule within the general umbrella of the collapse ...
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Confusion between the de Broglie wavelength of a particle and wave packets

So I learned that the de Broglie wavelength of a particle, $\lambda = \frac{h}{p}$, where h is Planck's constant and p is the momentum of the particle. I also learned that a quantum mechanics ...
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In interpretations of QM where the wave function is real, what does that mean?

In a lot of interpretations of Quantum Mechanics they believe that the wave function is "real". But what does that mean? Are they saying that the wave function of an elementary particle (electron/...
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Do quantum wave functions curve spacetime before they are measured

Do wave functions cause spacetime curvature before they are measured, or would curvature only happen upon measurement? I guess the question becomes, do quantum wavefunctions carry energy while they ...
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Orbital angular momentum of electrons

In a QM class, to study the hydrogen atom, we started by defining the Hamiltonian $H$ for a central potential, then made an orbital angular momentum operator appear as part of $H$, then down the line ...
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Projection of wavefunction onto basis function

I am given to believe that one way that one would could represent a wavefunction is by the expansion $$\Psi(x) = \Sigma_n \Psi_n(x) = \Sigma_n f_n\phi_n(x) \tag{1}$$ where $\{\phi_n (x) \}$ is an ...
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Orthogonality of summed wave functions

Problem. I know that the two wave functions $\Psi_1$ and $\Psi_2$ are all normalized and orthogonal. I now want to prove that this implies that $\Psi_3=\Psi_1+\Psi_2$ is orthogonal to $\Psi_4=\Psi_1-\...
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Correspondence between wave function and state vector

I am confused with connection between state $| \psi \rangle$ of a quantum system and corresponding wave function $\psi(x)$ (at a given time). I have been told that for every state $| \psi \rangle$ we ...
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Tip of a spreading wave-packet: asymptotics beyond all orders of a saddle point expansion

This is a technical question coming from mapping of an unrelated problem onto dynamics of a non-relativistic massive particle in 1+1 dimensions. This issue is with asymptotics dominated by a term ...