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Swapnil
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I don't get how the uncertainty principle is a direct consequence of wave-particle duality?
PhilosophyofPhysics said:If you assume that matter has a wavelike nature, then the uncertainty principle follows directly from the mathematics of waves.
Swapnil said:I don't get how the uncertainty principle is a direct consequence of wave-particle duality?
Why? If, with "strong" light beam you mean "with high intensity", then it's not so, because a light beam intensity is proportional to the number of photons.Confused2 said:Edit .. the same would seem to apply to a single photon DSE which I would expect to give the same result as a 'strong' light beam.
Confused2 said:This is based on an assumption that may be wrong ..
As far as I know the double slit experiment (with monochromatic light) gives perfect destructive interference which we can use to measure the wavelength of our light. Does the same experiment performed with mono-energetic electrons gives the same 'perfect' destructive interference?
If 'yes' then can we know the de Broglie wavelength of our electrons to any desired degree of accuracy? If yes again then doesn't this make our electron 'monochromatic' rather than the bunch of frequencies to be expected in a wavepacket?
Edit .. the same would seem to apply to a single photon DSE which I would expect to give the same result as a 'strong' light beam.
Confused2 said:If 'yes' then can we know the de Broglie wavelength of our electrons to any desired degree of accuracy?
ZapperZ said:Actually, here's something you may want to think about, and what I think many students just starting out in QM usually get confused with.
Solve, for example, the wavefunction for free particle with some energy. Now look at the wavefunction (you should be getting some plane waves solution). If you look, for example, in Marcella's full QM treatment of interference, you'll notice that such wavefunction (i.e. the one that you solved from the Schrodinger equation) is the one involved in the interference/diffraction/etc.
Now, is THAT the same thing as the "de Broglie wavefunction"? Remember that the de Broglie wavefunction is simply a function of the particle's momentum. It didn't care about the geometry of potential function that the particle is in. So when we shoot electrons at slits, or when we use them for experiments such as LEED, or when we have them as a supercurrent in a SQUID, did we use the de Broglie wavefunction for the calculations in obtaining all the interference/diffraction effects? Or did we actually use the wavefunction obtained from the Hamiltonian?
Zz.
Dense said:Isn't it sort of a definition of the Hamiltonian operator that it’s what you’d use to determine what happens to a waveform anyway? I'm new to all this, though, so I might not be following.
A free particle with a definite state of energy, which is moving through time and space, is going to have a waveform in a single plane of space.
You get that result if you start with a time-dependent Schrodinger equation, then solve separately for the time and space variables. You’re going to use the deBroglie wavelength, based on the relationship with momentum, to solve for position’s constant. You’re going to use the Planck hypothesis to solve for time’s constant. Then plugging those results in gets you a plane wave.
But that’s just for figuring out the kind of waveform that’s going to be undergoing diffraction and interference. Now that we know what kind of wave it is, we use the Hamiltonian to determine what’s going to happen to it. (Which gives you the Schrodinger equation all over again, doesn’t it?)
ZapperZ said:Er... say what? I can't follow a single thing that you said here.
Dense said:Doesn't surprise me.
Googling around after posting my garbled thought, I found this page http://hyperphysics.phy-astr.gsu.edu/hbase/quantum/scheq.html
The second box there says sort of what I was thinking, and more clearly.
ZapperZ said:Ah, now I understand what you're trying to do.
Note that the de Broglie "wave" was never used to derive the wavefunction of a free particle. That was the issue that I brought up earlier. However, when you try to relate the physical quantities of what you obtained out of the Schrodinger equation, only then do you make ad hoc use of the de Broglie relations.
So the wavefunction that you solve out of the Schrodinger Eq. isn't really the deBroglie "wave", even for a particle. We however make use of it to relate wave-particle quantites.
Zz.
Dense said:Ah! Yes! Thank you!
A light bulb just went on in my head. Thanks. (Now if I can just get the rest of them to turn on...)
The Uncertainty Principle, also known as Heisenberg's Uncertainty Principle, is a fundamental principle in quantum mechanics that states that the position and momentum of a particle cannot be simultaneously measured with perfect accuracy. This means that the more precisely we know the position of a particle, the less accurately we can know its momentum, and vice versa.
The Uncertainty Principle is closely related to the concept of Wave-Particle Duality, which states that particles can exhibit both wave-like and particle-like properties. This principle explains why it is impossible to know the exact position and momentum of a particle at the same time, as particles can behave like waves and their position becomes uncertain when they are in a state of superposition.
No, the Uncertainty Principle is a fundamental law of quantum mechanics and cannot be violated. It is a consequence of the wave-like nature of particles and is supported by numerous experimental evidence.
Although the Uncertainty Principle may seem abstract and only relevant in the microscopic world of atoms and particles, it actually has implications in our everyday lives. For example, it is the reason why we cannot predict the exact path of a hurricane or the exact location of an electron in an atom.
While the Uncertainty Principle cannot be violated, it can be mitigated through various techniques such as using more precise instruments or reducing the energy of the particle being measured. However, it is a fundamental principle of quantum mechanics and cannot be completely overcome.