So I'm reading about bound systems right now in my quantum text. It is beginning to explain why energy must be quantized, and is doing so by introducing the reader to the one dimensional "quanton in a box". Essentially it shows this:  For a region of 0 potential energy, with infinite potential energy everywhere else, a particle is "free" within the region of 0 potential energy. This means that the particles wave function must go to zero at these "boudaries", and therefore resemble a standing wave between the potential energy bounds.  Because the wavefunction must be zero at the ends, the Dr Broglie wavelength must be so that an integer number of half-wavelengths fits inbetween the regions of infinite potential. Therefore, the energy of a particle (quanton) must be so to satisfy this, and cannot be any arbitrary value.  First of all, the more board question: why, then would an electron that is outside of a bound system be quantized? Or any other particle that is outside of a bound system (atomic nucleus, etc.)?   Secondly, the question that I am more interested in: the only reason that a classical standing wave in a string results in the way it does is be cause f resonance; the string resists frequencies that do not match a natural harmonic of the material, and accepts very well frequencies which do match. So then isn't the fact of energy quantization a direct consequence of the phenomenon of resonance? It must be. But then, what is the material that is experiencing this resonance? For an electron, which could be described as a "matter-wave", would the Higgs field, itself, be resonating? Or something else?
21409
Given some potential $V$, we have the eigenvalue problem   $$ -\frac{\hbar^2}{2m}\Delta \psi + V\psi = E\psi $$   with the boundary condition   $$ \lim_{|x|\rightarrow \infty} \psi(x) = 0 $$   If we wish to seek solutions for   $$ E &lt; \lim_{|x|\rightarrow \infty}V(x), $$   I find textbooks like Sakurai and others assert that (without proof) set of all such eigenvalues $E$ that admits nontivial solutions is discrete, which gives us liberty to index them with natural numbers (principal quantum numbers). Can anyone help me with a little elaboration of this mathematical fact ? As a student of mathematics, I know that compact operators have discrete spectrums, so for the operator $ T : L^2 \rightarrow L^2 $given as  $$ T(\psi) = -\frac{\hbar^2}{2m}\Delta \psi + V\psi. $$   Are there methods to show that $T$ is compact? Or possibly other ways to establish discreteness of its eigenspectrum?
41100
I am trying to understand intuitively what a  is, but for the moment I find it quite difficult (having a limited background in quantum mechanics, an undergraduate course in non-relativistic QM). In fact, I find it hard to formulate good questions, so I hope my questions below make some sense.  I read that phonons are (the quantum mechanical analog of) normal modes of vibration in a crystalline system of atoms or molecules, so I guess a superposition, i.e. a general vibration should also be a phonon. Is that so? Why would they then be described as normal modes?  Could we say that a phonon is a particle whose position wave function extends over the whole crystal? Are the quantum mechanical frequency and wave vector the same as the frequency and wave vector of the corresponding classical oscillation (vibration in the crystal)?  In what sense is it (like) a particle? In that it is always observed or it always interacts at a specific location?
8845
Srednicki's Quantum Field Theory mentions the following at the end of the unit on path integrals in non-relativistic quantum mechanics:   Assume that the total Hamiltonian is of the form,     $$ H = H_0 + H_1, $$   where $H_0$ is the exactly solvable part and $H_1$ is treated as a perturbation.  Without the presence of the perturbation, the ground-state t0 ground-state transition amplitude in the presence of external forces/classical sources is:     $$ \langle 0 | 0 \rangle_{f,h} = \int \mathcal{D}p \; \mathcal{D}q \; \exp \Bigg [i \int_{-\infty}^{\infty} dt \; (p\dot{q} - (1- i\epsilon)H_{0} + fq + h p \Bigg ]. \tag{6.21} $$   In the presence of the perurbation, the amplitude is now given by:    Questions:  1) To get equation $(6.22)$, how can we take the exponential inside the integral, make the necessary manipulations and then take the exponential outside the integral. This seems very ad-hoc to me. How and when do we know that the these steps are valid? Doesn't this required justification from mathematical analysis?  2) More importantly, I don't get the first term involving the perturbing Hamiltonian. Why do we have functional derivatives in its argument? How did it come out to be? Why don't we similarly have functional derivatives in the argument of $H_0$ as well? What's the distinction?  Note: The factor of $(1 - i \epsilon)$ has been suppressed in $(6.22)$.
81369
I was recently reading Griffiths' Introduction to Quantum Mechanics, and I stuck upon a following sentence:     but $\Psi$ must go to zero as $x$ goes to $\pm\infty$ - otherwise the wave function would not be normalizable.   The author also added a footer: "A good mathematician can supply you with pathological counterexamples, but they do not arise in physics (...)".  Can anybody give such a counterexample?
45265
I want to ask if it is reasonable that I use the Dirac-Delta function as an initial state ($\Psi (x,0) $) for the free particle wavefunction and interpret it such that I say that the particle is exactly at x=0 during time t=0? If I use this initial state, can I also use it to predict how the wavefunction should evolve in time? That is, if $\Psi (x,0) = \delta(x) $, then, $$ \phi(k) = \frac{1}{\sqrt{2 \pi}} \int_{-\infty}^{\infty} \delta(x)e^{-ikx} dx=\frac{1}{\sqrt{2 \pi}} $$ then, $$ \Psi(x,t)= \frac{1}{\sqrt{2 \pi}} \int_{-\infty}^{\infty} \phi(k)e^{i(kx-\omega t)} dk $$ or, $$ \Psi(x,t)= \frac{1}{2 \pi} \int_{-\infty}^{\infty}e^{i(kx-\omega t)} dk. $$  Is the above wavefunction a valid expression to see how a particle that is initially localized in the origin evolves in time?
20186
I am totally a beginner in QFT, here's the problem that I got: for the real scalar fields, are there any elementary particles descriped by them. If so, how to understand the negative probability problem?
94335
In physics there is quanta and energy occurs per this unit. Is it it then reasonable that probability also is quantized since energy is?
24167
First, I'll state some background that lead me to the question.  I was thinking about quantization of space-time on and off for a long time but I never really looked into it any deeper (mainly because I am not yet quite fluent in string theory). But the recent discussion about  got me thinking about the topic again. Combined with the fact that it has been more or less decided that , I decided I should give it a go.    So, first some of my (certainly very naive) thoughts.  It's no problem to quantize gravitational waves on a curved background. They don't differ much from any other particles we know. But what if we want the background itself to change in response to movement of the matter and quantize these processes? Then I would imagine that space-time itself is built from tiny particles (call them space-timeons) that interact by means of exchanging gravitons. I drawed this conclusion from an analogy with how solid matter is built from atoms in a lattice and interact by means of exchanging phonons.  Now, I am aware that the above picture is completely naive but I think it must in some sense also be correct. That is, if there exists any reasonable notion of quantum gravity, it has to look something like that (at least on the level of QFT).  So, having come this far I decided I should not stop. So let's move one step further and assume string theory is correct description of the nature. Then all the particles above are actually strings. So I decided that space-time must probably arise as condensation of huge number of strings. Now does this make any sense at all? To make it more precise (and also to ask something in case it doesn't make a sense), I have two questions:        In what way does the classical space-time arise as a limit in the string theory? Try to give a clear, conceptual description of this process. I don't mind some equations, but I want mainly ideas.   Is there a good introduction into this subject? If so, what is it? If not, where else can I learn about this stuff?      Regarding 2., I would prefer something not too advanced. But to give you an idea of what kind of literature might be appropriate for my level: I already now something about classical strings, how to quantize them (in different quantization schemes) and some bits about the role of CFT on the worldsheets. Also I have a qualitative overview of different types of string theories and also a little quantitative knowledge about moduli spaces of Calabi-Yau manifolds and their properties.
73280
Max Planck, arguably the leading theoretical Physicist of his day, apparently used of experimental data to formulate his equation for , which was a major step in the birth of quantum mechanics. For this seminal event of the 20th century, I would expect a detailed explanation of how Planck arrived at his conclusion that energy MUST be quantized. I've even read his Nobel lecture but was non the wiser.  Can some kind person please explain it, or give me even just a hint?
78641
In deriving the quantization condition for a bound state in a potential with "no verticle walls" we start with the WKB connection formulas to find the wavefunction in the interior of the well ($x_1&lt;x&lt;x_2$), namely $$\psi(x) \cong \begin{cases}\displaystyle\frac{2C_1}{\sqrt{p(x)}}\sin\left[\frac{1}{\hbar}\int_{x_1}^x{dx\ p(x)}+\frac{\pi}{4}\right]&amp;\quad\text{for}\quad x_1&lt;x\\ \\\displaystyle \frac{2C_2}{\sqrt{p(x)}}\sin\left[\frac{1}{\hbar}\int_x^{x_2}{dx\ p(x)}+\frac{\pi}{4}\right]&amp;\quad\text{for}\quad x&lt;x_2 \end{cases}$$  Our task is to connect the two wave functions in the interior region together. Seeing that the coefficients are easily handled, it is then necessary that the arguments of the sine functions be equivalent. Denoting  $$\theta_1(x) = -\frac{1}{\hbar}\int_{x_1}^x{dx\ p(x)}-\frac{\pi}{4}$$  $$\theta_2(x) = \frac{1}{\hbar}\int_{x}^{x_2}{dx\ p(x)}+\frac{\pi}{4}$$ we have  $$\theta_1(x)+n\pi = \theta_2(x)\quad\text{for}\quad n = 0,1,2,3..$$  since  the relative minus can be absorbed into either coefficient. Thus $$-\frac{1}{\hbar}\int_{x_1}^x{dx\ p(x)}-\frac{\pi}{4}+n\pi= \frac{1}{\hbar}\int_{x}^{x_2}{dx\ p(x)}+\frac{\pi}{4}$$ \begin{equation}\int_{x_1}^{x_2}{dx\ p(x)} = \left(n-\frac{1}{2}\right)\pi\hbar\quad\text{for}\quad n=0,1,2,3..\end{equation} This is our quantization condition for the allowed energies.   However, following the derivation in Griffiths or Sakurai, $n=0$ is not included. Why?   My ideas: Well this would imply  $$\theta_1(x) = \theta_2(x)$$ \begin{equation}\int_{x_1}^{x_2}{dx\ p(x)} = -\frac{\pi}{2}\hbar\end{equation} \begin{equation}\int_{x_1}^{x_2}{dx\ \sqrt{2m(E-V(x))}} = -\frac{\pi}{2}\hbar\end{equation} I am trying to imagine why a negative value of the action integral might be prohibited. Am I correct in assuming that if we were to restrict all possible values of the energy to be real, it would that imply $$p(x)&gt;0\ $$ Can someone provide a proof as to why  $$p(x)&gt;0\rightarrow E\in\mathbb{R}. $$It would then follow the the integral of a strictly positive integrand must itself be positive, which means we cannot have $-\frac{\pi}{2}\hbar$.
112152
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 do with density matrices?    Please try to give a mathematical explanation that is simple and concise...
56669
So I've been reading a bit and watching some videos about the double slit experiment, and therefore the wave particle duality; I've also read this "implies" that a particle can be in two places at the same time. Can you clarify this, and how could that be possible?
4897
I never learnt QFT and I apologize for my (probably) elementary question. Somebody told me that in QFT a particle is viewed as an irregularity in the field.  On the other hand, in an  in Wikipedia I see the sentence "A QFT treats particles as excited states of an underlying physical field, so these are called field quanta."  Which one of the true is a better description? The 1st description hints that the particle is a localized phenomenon inside a field that maybe occupies a big region in space. The 2nd description speaks of an "underlying" field. So, is there a field and in addition there is a particle? If it is, then what is the occupation number of that "underlying" field?  None of these approaches is clear to me, I know the approach in QM, and none of them resembles the QM.  The motivation behind my question is a certain similarity that I find between the above descriptions and the Bohm interpretation of QM, (i.e. the background field - in Bohm's interpretation there is a background quantum potential - and a particle floating in it.)  In all, is a particle treated in QFT as a localized phenomenon inside a field occupying a wider volume? I would appreciate a simple and direct answer.
121205
According to various sources on the web, it seems like the general concensus is that there isn't actually any wave-particle duality with quantum particles. For example,  implies that diffraction patterns in  were interpreted as wave interference due to apparatus limitation at the time they were first performed.  Does this mean that all those sources and animations showing two waves interfering are simply incorrect, classical conclusions which don't have anything to do with (quantum) reality?  What's actually the most confusing is that most sites which state that it is now possible to pass individual photons through these slits, also claim that these individual photons somehow interfere with themselves resulting in the observed patterns. That seems like a rather thin explanation, doesn't it?  So, is there actually any need to use wave interference to explain the phenomena, or can we simply state that the pattern is probabilistic in a certain way, without involving the "spooky" explanations?
118416
Yesterday I asked this  concerning the existence of terms. This question is on the same topic. I have been looking at some tables (in Yang and Hamilton (2010), pg 199) at the terms that exist for two electrons in the same configuration. I noticed that for example if the state $^1S$ exists the state $^3S$ would not and vice versa. Is this a general rule? I.e. can we say:     For two electrons in the configuration $nl^2$ if the term $^1L$ exists then the term $^3L$ will not and vice versa.   If so can it be proved easily and if not why not?
50368
The quantum mechanics of Coloumb-force bound states of atomic nuclei and electrons lead to the extremely rich theory of molecules. In particular, I think the richness of the theory is related to the large mass ratio between the nucleon and the electron. This mass ratio leads to the Born-Oppenheimer approximation which gives rise to a complicated effective potential for the nuclei which posses many local minima: the molecules.  I wonder whether there are analogical phenomena in which the bound states are gravitational. It seems that if we take a collection of electrically neutral molecules and neutrinos, it should be possible to form a large number of bound states, in particular because of the large (not so long ago deemed infinite) mass ratio between molecules and neutrions. Of course neutrinos are highly relativistic and I can't tell how it affects things.  Now, even if we leave neutrinos alone, the typical size of such a bound state is  $$\frac{\hbar^2}{G  m^3}$$  where $m$ is proton mass. Google calculator reveals this to be 3.8 million light years. Holy moly! However, this is still much smaller than the observable universe. Can there be places in the universe sufficiently empty to contain such bound states? What would be the effect of general relativistic phenomena (expansion of space)? EDIT: I guess no place is sufficiently empty at least because background radiation is everywhere. Maybe these creatures will become relevant in a very distant future when the background radiation cools off considerably?  Summing up:     What is known about the quantum mechanics of gravitational bound states of electrically neutral molecules and neutrinos?   I'm tagging the question as "quantum gravity" since it involves gravity and quantum mechanics. Of course it is not quantum gravity in the usual sense of studying Planck-scale phenomena. I think the tag is still appropriate  EDIT: Gravitional bound states of molecules will often be unstable with respect to collapse to a van der Waals bound state (thx Vladimir for bring up the issue of vad der Waals interaction in the comments). However the lifetime of these states is very long.
39100
I'm aware that there're some questions posted here with respect to this subject on this site, but I still want to make sure, is frequency quantized? Do very fine discontinuities exist in a continuous spectrum like the black body spectrum?