People often wonder why imaginary numbers show up in quantum mechanics. And one approach is to just say "well we need them for these derivatives to work out". But drawing the similarity to Fourier pairs brings them in fairly naturally. I just find approach so neat! I have a bit of an unrelated question. You mention at one point that this is very non-relativistic because the Schödinger Equation treats space and time separate. Sean Carroll is very adamant that the SE is actually perfectly relativistic and all you need for QFT (or ultimately all of physics, really). I'm assuming he means the time-independent SE and we basically stick relativity into the Hamiltonian? Or can we rewrite it using the d'Alambert operator instead of Laplace to make it relativistic?
@Narf Whals, this is a great question and a great point. Looking back, I think I was not as precise as I should have been. What I think Carroll is referring to is that, to write down a QFT, we do only need the i*hbar*dPsi/dt = H Psi form of the Schoedinger equation. One way to see that this can fit into special relativity is that if we just define H to be the zero-component of the 4-momentum operator, then we can see that the 4-momentum generates translations of the 4-dimensional spacetime vector with x^0 = t. However, the issue is that a lot is hidden in "H," and the form of H that we have been using is absolutely not suitable for a relativistic quantum theory. This can easily be seen since the energy of a free relativistic particle is Sqrt(p^2 c^2 + m^2 c^4) unlike our classical p^2/(2m). The square root makes things tricky since it isn't well-suited for the linear properties that we want when we upgrade the momentum to a momentum operator. There are sort of two ways around this: First, we can try to "square" both sides, in which case we end up with the Klein-Gordon equation. The problem with this is that it results in negative-norm states, so we can't interpret Psi^* Psi as a probability density and it is very tricky to figure out what this is actually telling us (also, it doesn't account for spin-1/2 particles) The other option is to use a Hamiltonian which is naturally relativistically invariant, even within Newtonian mechanics. This happens to be a property of many field equations, but the issue here is that we have to replace our position and momentum operators with corresponding field operators. This is what is known as canonical quantization. So hopefully that gives a bit more insight into this whole idea of why we need fields to do relativistic QM!
@@zapphysics I'm actually not entirely sure what the difference is between the wave function and the field. What's different from an operator acting on Psi and an operator acting on a field? Aren't the operators still x and -i hbar dx (in position space)?
Respected sir 🌟, consider we have an Stern gerlach DETECTOR oriented right to left , IF WE PASS AN SINGLE SPIN UP ELECTRON (detected UP by another detector) it would be either DEFLECTED RIGHT OR LEFT in our detector....IF WE DO NOT OBSERVE IT AND MAKE IT TO PASS THROUGH....EVEN THOUGHT WE DO NOT OBSERVE....THE GERLACH DETECTOR WOULD EXPERIENCE AN NET FORCE OPPOSITE TO THE DEFLECTION DIRECTION OF ELECTRONS(Newton's law) RIGHT?? IS THAT AN MEASUREMENT??? DOES IT COLLAPSES THE WAVE FUNCTION??? Does gerlach DETECTOR always MEASURES while DETECTING?? THANK YOU sir 🌟🌟
Respected sir🌟🌟, During interference of EM wave... electric field adds up like vector....if two electric field of EM wave of same amplitude ,frequency,inphase...when added by vector gives,E+E= 2E, but when we add Energy of E field WHICH IS PROPORTIONAL TO E^2...it contradicts (€E^2=2€E^2).... Similarly for distructive interference of two similar PLANE Em wave of PHASE DIFFERENCE of π interfering DESTRUCTIVELY COMPLETELY... IF E field becomes ZERO EVERY WHERE..... WHERE DOES ITS ENERGY GO?what is wrong here? Thank you sir 🌟
All his previous streams on classical mechanics are in this playlist. The one he refers to in the beginning is the last one(video 16) ru-vid.com/group/PL-RmwJq2kMwmQN7hmbfmmJEJLqGhQWe1x
Respected sir 🌟, will everyday objects such as a ball When passed through a double slit experiment in space where there is no light and air (NO INTERACTION) so no measurement...will ball produce an inference pattern on the screen? Thank you sir 🌟
@gowri ssshanker the thing that you have to keep in mind here is that everyday objects aren't just single, isolated things. They are really made up of billions upon billions of atoms (each of which are made up of their own, more fundamental constituents). Each of these particles will get their own wavefunction and since all of these particles are constantly interacting, the collective "quantum-ness" of everything combined essentially cancels out. This is why we don't tend to see quantum effects on large scales other than very special cases like a Bose-Einstein condensate or superfluids or things like that. However, people have still seen double-slit interference with relatively large objects. If I'm not mistaken, they can now see this interference with C-60 "buckyball" molecules which are relatively huge compared to, say, electrons or neutrons which in my opinion is quite amazing!
