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 discontinuous when there is a Dirac delta potential, but I can't see what is forcing the wavefunction to be continuous in the case of an infinite potential (mathematically as well as physically).
Can someone explain this to me?
I regularly find that I'll understand where the field content in a particular physics paper comes from, but then a Lagrangian or action or superpotential is stated and I don't know how it's derived. Is there a set of general rules for building a Lagrangian/action/superpotential if you already know the field content of the theory?
Any suggestions of sources that explain how to do so would be very welcome as I'm having little trouble finding much that helps.
Answer
Building an action: If you know the field content (which I assume means you know the gauge group and reps of all the fields) then:
Write down every term that is Lorentz scalar (so combinations like $\partial_\mu A^\mu$, $\bar{\psi}\gamma^\mu \partial_\mu\psi$ allowed but not things like $\vec{n}\cdot\nabla \phi$ where $\vec{n}$ is some random 3-vector). Stop at terms with dimension greater than the spacetime dimension (4 in 4D, so include terms like $\phi^4$ and $\phi\bar{\psi}\psi$ but not higher dimension terms like $\phi \partial_\mu \phi \bar{\psi} \gamma^\mu \psi$).
Cross out terms that are not gauge invariant. This means that gauge fields can only appear through covariant derivatives and field strength tensors, and matter fields must appear in singlet combinations.
Cross out terms that violate any global symmetries you want to impose (though be warned - if these symmetries are anomalous you can't drop these terms consistently). In SUSY theories you need to impose relations between coupling constants as well.
After you've done this you can probably use field redefinitions (orthogonal/unitary rotations in flavour space) to simplify some of the coupling constants. An example would be the standard model where the lepton yukawas can be diagonalised and the quark yukawas can be diagonalised leaving just the physical CKM matrix.
It's really a lot like lego - the fields are the building blocks and symmetries & gauge invariance tell you what you can put together.
I understand that the closer something travels to the speed of light, that time will stretch by a factor, and distance will compress by the same factor.
My question is, if something travels in a circle, close to the speed of light, what does the distance of the journey look like to them? They measure that the trip took them 10 minutes. And an outside observer says that the journey took 20 minutes, and the outside observer measured that they did, for example, 1000 laps of a circle circumference 1000 km.
So if the plane had a distance trip counter, what would it read? And if they were looking out of the window, would the circle still look like it had a circumference of 1000km?
Answer
Circular motion in special relativity is somewhat tricky: Note that for circular motion, the acceleration in the spaceship travelling in a circle is not zero, so the spaceship is not in a single frame of inertia.
Here is an interesting thought: Distances perpendicular to the direction of motion are not subject to contraction. Hence, if the observes on earth see the spaceship going on a circle with radius $R$, then in their frame of inertia the spaceship is always a distance $R$ away from the earth. Since the line from earth to spaceship is perpendicular to the direction of flight, the people on the spaceship will also believe that they are always a distance $R$ away from earth, so they will also fly on a circle.
They will, nonetheless, experience a different circumference! The best way to solve this is to consider a polygon with $N$ sides and then let $N$ go to infinity. If people on the earth measure each side of the polygon as $L_0/N$ where $L_0$ is the circumference of the polygon in the earth's frame of inertia, then the spaceship-people will measure each side to be $L_0/(N\gamma)$. Hence, for $N \rightarrow \infty$, the polygon becomes a circle. Measured from earth, it has circumference $L_0$, but for the spaceship it has circumference $L_0/\gamma$.
This suggests that the spaceship moves through non-Euclidean geometry, because it travels on a circle whose ratio between circumference and diameter is less than $2\pi$. This is a hint that accelerating frames have non-Euclidean geometry, which is excessively treated in General Relativity.
Reference: http://abacus.bates.edu/~msemon/WortelMalinSemon.pdf
A stranger walked up to my friend and drew these symbols on his arm with a Sharpie. I have no idea what they mean.
They were written on his right and left arm. The second picture might be upside down.
I imagine that they are based upon fictional languages. If so, what languages, and what do these messages read?
I ran across a neat quantum mechanics question recently:
Consider a scattering gedankenexperiment in which spinless particles of a given energy are directed at a target. We wish to measure the wave function downstream from the scatter. We assume the beam is described by a pure state, and that the incident wave is a plane wave (over a sufficiently large spatial region). The beam is low density, so the particles do not interact with one another. To measure $|\psi(r)|^2$ over some volume of space, we just put a screen $S$ in a certain location, and gather enough statistics to get the probability density on this surface. We then move the surface and measure again.
Describe a modification by which the phase of the wave function $\psi(r)$ can be measured on the screen, apart from the overall phase, which is nonphysical and can never be measured.
I imagine it can be possible to infer the phase by seeing how the scattered wavefunction interferes with a "reference" wave, which can be done by, e.g. splitting off a piece of the incoming wave and routing it around the target to the desired point. But this feels a bit convoluted. Is there a nice and direct way to measure the phase of the scattered wave?
Answer
I'll describe here a conceptual method for phase measurement without going into the details of the experimental complications, the conditions of validity, or the accuracy.
Given a system of particles of mass $m$ described by the wave function in a stationary state. $$\psi(\mathbf{r}) = A(\mathbf{r}) e^{i\phi(\mathbf{r}) }$$ The probability current is given by: $$\mathbf{J} = \frac{\hbar}{2 mi } (\psi^{\dagger} \mathbf{\nabla} \psi - \psi \mathbf{\nabla} \psi^{\dagger} ) = \frac{\hbar}{ m } A(\mathbf{r}) \mathbf{\nabla} \phi(\mathbf{r})$$ Suppose that you have very small directional sensors, which can measure the intensity perpendicular to their surface, and a very precise timer. Then in principle you can put sensors along the coordinate axes in the vicinity of some point in space, and measure the currents which will be proportional to $\frac{\partial\phi}{\partial x}$, $\frac{\partial\phi}{\partial y}$, $\frac{\partial\phi}{\partial z}$. These measurements will allow the construction of the phase up to an additive constant.
The measurement results will be proportional to the square root of the intensity: $\sqrt{|A|^2}$; and in order to remove this factor, we will need an additional volumetric sensor to measure the intensity in a small volume around the same point in space.
This principle is the basis of the widely known noninterferrometric phase measurement (or phase retrieval\reconstruction) techniques.
I have found the following interesting article: http://arxiv.org/abs/0706.0924
The authors examine the radial momentum operator in detail, in particular its time evolution due to the forces acting on the electron in the Hydrogen atom. Their key result is equations (31), (32) and (33). Here they derive explicit expressions for the centripetal force (resulting from the Coulomb interaction between electron and nucleus) and the centrifugal force (which appears to be the result of conservation of angular momentum).
The authors then demonstrate that, upon averaging, the effects of the centripetal force and the centrifugal force become identical. Hence on average there is no force, and therefore the radial momentum remains constant -- precisely as expected for a time-independent state.
Unfortunately the authors do not comment on the special case of the Hydrogen ground state (where $N=1$ and $L=0$), in which case there is no angular momentum. So how can there be a centrifugal force associated with it?
The formulas by the authors are in terms of a numerator and a denominator, both of which become zero for $L=0$. The quotient of numerator and denominator is tacitly assumed to yield the same (constant) value as in the more general case $L > 0$.
My questions: Is there indeed a centrifugal force acting on the electron in the Hydrogen atom, which on average balances the centripetal Coulomb force? If so, how can we understand this force in the case of $L=0$, given that the only obvious source (= angular momentum) equals zero in this particular case?
I have just started reading nuclear physics.I know that the sum of masses of the quarks is less than the proton or neutron itself as a whole . But why is it that the sum of the masses of the nucleons(protons and neutrons) is more than the nucleus itself? whats the difference between the two cases?
