### gravity - Did Einstein propose a perpetual motion machine to try to disprove quantum mechanics?

In response to quantum mechanics, so the story goes, Einstein proposed a machine, that, based on the uncertainty principle, was a perpetual motion. This showed that quantum mechanics was at odds with evidence that energy is conserved. Bohr later showed that the analysis was flawed; Einstein had failed to take gravity into account (ironically).Is this story true? If so, what was the machine Einstein proposed, how was it supposed to work, and what did Bohr reveal about it?...Read more

### quantum mechanics - Historically how it was discovered that we need fields to describe matter?

This question is from one historical perspective. The question is: how physicists historically found out that one needs quantum fields to describe matter?Being more detailed. Let us consider the electromagnetic field for a while. Classically this was already a field. Now, if I understood the history correctly, in the days of old quantum theory, when Planck proposed the solution to the blackbody radiation problem in terms of quantized energy levels, and when Einstein did the same to solve the photoelectric effect problem, they were essentialy pr...Read more

### quantum mechanics - 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, $$\hat{H}=\hat{T}(\hat{p})+\hat{V}(\hat{x}).$$ I mean the sum of Kinetic operator and potential operator term. Can I have Hamiltonian without a kinetic term, $$\hat{H}=\hat{x}^{2}$$ or in other words? Can I have a Hamiltonian with just potential term alone? $$\hat{H}=\hat{V}(\hat{x}).$$Does quantum mechanics allow that?...Read more

### quantum mechanics - The problem of a relativistic path integral

Many books have described the path integral for non-relativistic quantum. For example, how to get the Schrödinger equation from the path integral. But no one told us the relativistic version. In fact, the relativistic version is impossible to be perfect, it must be replaced by quantum field theory. But why?The answer I want is not that the quantum mechanics will give us a negative energy or negative probability. We need a answer to explain why non-relativistic Lagrangian $$L=\frac{p^2}{2m}$$ can lead a correct Schrödinger equation? Why if we re...Read more

### quantum mechanics - Refraction and how light bends

I have heard a particle nature explanation of how light continues to go with the same constant speed $c$ after it has passed through a denser medium. I also have come across how photon is absorbed by the dielectric molecules and then again re-emitted after a fleeting period of $10^-15$ seconds and that is how light is able to continue in its constant speed condition.My questions are,How does light bend at the interface of the two media? Could you please give an explanation without referring to the wave nature of light?(I know the Fermat's princ...Read more

### quantum mechanics - If a theory gets two predictions right, how likely it is that the rest of the predictions are true too?

The question lucidly defines what I am trying to inquire, so there is no need to elucidate it any further. Another question would be, General/Special Relativity has gotten some predictions right as experimental evidence is backing it up, but is there anything that it got wrong?...Read more

### reference frames - Quantum mechanical origin of pseudo forces

I am thinking about this from quite some time but could not come up with any satisfactory explanation. In a nutshell, how would one explain the pseudo forces felt by non-inertial observers given that the fundamental laws of physics are quantum mechanical? Since in quantum mechanics one always talks about potentials instead of forces, I cannot think of anything that I can relate to the acceleration. In other words, given an electron for example, can we say that in the frame of an electron there is exists a pseudo force? I think no because of cou...Read more

### quantum mechanics - What is an analog to QM's Hilbert space in GR?

I've read that QM operates in a Hilbert space (where the state functions live). I don't know if its meaningful to ask such a question, what are the answers to an analogous questions on GR and Newtonian gravity?...Read more

### energy conservation - Quantum perpetual motion

Perpetual motion describes hypothetical machines that operate or produce useful work indefinitely and, more generally, hypothetical machines that produce more work or energy than they consume, whether they might operate indefinitely or not.(Source:Wikipedia)With this definition in mind, particularly the "operates indefinitely" (I don't care about producing work), won't quantum mechanics allow perpetual motion due to energy quantization?For example, an electron in hydrogen can be thought of as perpetual motion. It's indefinite(I think so); unlik...Read more

### quantum mechanics - What's the relationship between Density Functional Theory (DFT) and Kohn-Sham equations?

It seems Kohn-Sham equations are approximate methods to solve many body Schrodinger's equation. They directly split a multi-electron Schrodinger equation into many single-electron Schrodinger equations with exchange-correlation energy to eliminate errors.Why there is a necessity to introduce Density Functional Theory before give Kohn-Sham equations? Or if there is no Density Functional Theory, can the Kohn-Sham equations still be used?...Read more

### quantum mechanics - How to integrate two-electron interaction in He? (variational method)

A probe wavefunction in the variational method is $$\psi(r_1, r_2) =\frac{\alpha^6}{\pi^2}e^{-\alpha(r_1+r_2)}$$. In $\left<\psi \right|H\left|\psi\right>$ with $$H = \frac{p_1^2+p_2^2}{2m} - \frac{Ze^2}{r_1}-\frac{Ze^2}{r_2}+\frac{e^2}{|\vec{r_1}-\vec{r_2}|}$$the last term is to be integrated like that: $$\idotsint_{} \frac{\left|\psi\right|^2 e^2}{|\vec{r_1}-\vec{r_2}|}r_1^2\, r_2^2\,\sin{\theta_1}\sin{\theta_2}\, d\theta_1 d\theta_2\,d \phi_1d\phi_2\,dr_1dr_2,$$which is quite challenging for me. Does anyone know how to integrate it or...Read more

### quantum mechanics - Struggling with an integral

I'm struggling with the following integral:$$\int \int (r_1^2 + r_2^2) \exp \left( -\frac{b (r_1 + r_2)}{a} \right) \, \mathrm{d}V_1 \, \mathrm{d}V_2$$I tried to expand near $r_1 = 0 ;\; r_2 = 0$ and to move to s spherical coordinates, but can't get through. I thought there might be some trick I have forgotten to evaluate integrals like this.It is related to a diamagnetic susceptibility of the helium atoms....Read more

### quantum mechanics - How to choose proper measurement operator?

Let's assume I have two states inside the Bloch sphere, at radial vectors $r_1$ and $r_2$ respectively $(r_1<r_2<1)$. Their angular location is same. These are like:$$\rho = \begin{pmatrix}\frac{1+r_1 \cos\theta}{2} &\frac{r_1 \exp(-i\phi)\sin\theta}{2} \\\frac{r_1 \exp(i\phi)\sin\theta}{2} &\frac{1-r_1 \cos\theta}{2} \nonumber\end{pmatrix}$$and another state as \rho' = \begin{pmatrix}\frac{1+r_2 \cos\theta}{2} &\frac{r_2 \exp(-i\phi)\sin\theta}{2} \\\frac{r_2 \exp(i\phi)\sin\theta}{2...Read more

### quantum mechanics - Minimizing the energy of a Slater determinant: why are the Lagrange multiplier elements of a Hermitian matrix?

If I want to minimize the energy of a Slater determinant subject to the constraint that the spin orbitals are orthonormal (as in the Hartree-Fock approximation), I can use Lagrange's method of undetermined multiplier, i.e.$$L[\{\chi_{a}\}] = E_{0}[\{\chi_{a}\}]-\sum_{a=1}^{N}\sum_{b=1}^{N}\varepsilon_{ba}([a|b]-\delta_{ab})$$where $\{\chi_{a}\}$ are the spin orbitals, $E_{0}$ is the ground state energy, $[a|b]$ is the overlap integral between spin-orbitals $\chi_{a}$ and $\chi_{b}$ and $\varepsilon_{ba}$ is a Langrange multiplier. By setting t...Read more