The wave in quantum mechanics is not a wave in true sense of water wave. It is a mathematiccial wave. It is a probability wave. I think I get till here. The question is does this probability wave (for example probability of finding a particle over some region) vary smoothly like a classical waves? For example a classical wave have some pattern and that pattern repeats. There is certain order in that wave. These probability waves do they have some order, pattern like classical waves or can this wave change abrubtly?
Asked
Active
Viewed 303 times
-2
-
3Just to be more exact, you are essentially asking if $\psi(x)$ and $\psi'(x)$ have to be continuous? – BioPhysicist Nov 03 '18 at 03:43
-
1Are you asking the properties of wave: 1) reflection, 2) refraction, 3) diffraction, and 4) interference ? – K_inverse Nov 03 '18 at 04:41
-
2Possible duplicates: https://physics.stackexchange.com/q/19667/2451 , https://physics.stackexchange.com/q/1067/2451 and links therein. – Qmechanic Nov 03 '18 at 06:51
-
5"For example a classical wave have some pattern and that pattern repeats" - is this true? Consider, for example, a shock wave. – Alfred Centauri Nov 03 '18 at 10:41
-
4or a single propagating pulse – user45664 Nov 03 '18 at 16:51
3 Answers
1
Actually, one of the most basics and fundamental assumptions in common orthodox quantum mechanics is that the space and the wave function is smooth. So, that takes cares of what mathematicans would mean by continuous and differentiable. Mathematically, there aren't many difference between a quantum wave and a classical wave, just one is in complex space(so there is path integration, imaginary unit etc. which seemed different), the other one is in the real space. In short, they are both waves, that's why textbooks usually teach quantum waves started from classical waves.
As far as one can describe the quantum system with the wave function, you can think it as a classical wave distribution in the sense that $\frac{d}{dt}(\psi^*(x)\psi(x))$ is "smooth". (In fact, you can talk about Fourier transforms of quantum waves and the Fourier transform of a classical wave. In that sense, in frequency space, there really is not too much difference.)
But again, quantum is quantum, and complex waves are complex waves. It's much more generalized than classical waves, and notice that 2 dimensional complex space is isomorphic (as a vector space) to 4 dimensional real space, and the field of complex numbers is not an ordered field etc.
0
As Aaron Stevens hinted at, if you are asking:
'are $\psi(x)$ and $\psi'(x)$ continuous?'
Then the answer depends on the boundary conditions. If the boundary is not infinite then both the wavefunction and its first derivative will be continuous. However, at an infinite boundary you will find a discontinuity in the derivative. There is a nice derivation here: https://quantummechanics.ucsd.edu/ph130a/130_notes/node141.html
TanyaR
- 401
- 2
- 12
-2
There is no mystery at atomic level, it is how we describe system. Diffraction from slit is an example of classical mechanics, where distribution of point source is continuous, so intensity at a point is multiple of number of points and strength of individual point.
Quantum mechanics is interference of many points, diffraction grating and there is minimum space where no source can be located. This is forbidden region. The intensity at a point is multiple of square of number of points and strength of individual point.
This is in space, examples of this difference in time is Michelson interferometer and Fabrey-Perot interferometer. So it is how we arrange things or how we look at them. Due to many particles, there is always non-instantaneous effects.
