Uncertainty Principle cause infinite wavefunction solutions?

[a dull boy]

Dear Physics Forum,

Is the Uncertainty Principle the cause of the infinite solutions to Schrödinger's equation? I get the sense it is not. Could you elaborate a little?

Thanks, Mark

 

[Nugatory]

No, completely unrelated.

But maybe first you should think about why you would expect anything other than multiple solutions to Schrödinger's equation. Does it surprise you that classical mechanics says that more than one pattern of waves on the surface of the ocean is possible, and that this pattern can change over time as the waves move, hit land, are disturbed by passing ships? Every one of these patterns, at any moment and over the surface of the entire earth, is a different solution to the classical equation that describes the movement of water.

The movement of a particle is described by Schrödinger's equation. Particles can move in different directions at different speeds, so you'd expect to find multiple solutions to Schrödinger's equation.

 

[a dull boy]

...I think part of my confusion has to do with Einstein's comment that God does not play dice. I can see two source of this comment, the Uncertainty Principle and the infinite wavefunction solutions. For the latter, I thought there was only a single wavefunction solution for each principle quantum number n. But I am pretty sure I am wrong, and psi squared gives you the probability of the particle being at the Bohr radius for a particular energy level. For a given energy level, why is it only probable the electron is at the Bohr radius? I thought perhaps the Uncertainty Principle, but I'm wrong.

Thanks for your help Nugatory

 

[a dull boy]

One more followup, I was speaking about stationary states, and my sense was that you got infinite solutions for a single particle speed/momentum...

 

[Nugatory]

...I think part of my confusion has to do with Einstein's comment that God does not play dice. I can see two source of this comment, the Uncertainty Principle and the infinite wavefunction solutions. For the latter, I thought there was only a single wavefunction solution for each principle quantum number n.

(Actually there are multiple solutions for each principal quantum number, with different values of angular momentum and spin (represented by the m, l, and s numbers); this is just a property of the particular forces acting on the electron in this particular situation. However, that's beside the point here; the multiple solutions for the same n aren't the source of the randomness to which Einstein objected.)

The probabilistic nature of quantum mechanics comes from the fact that if $\psi_1$ and $\psi_2$ are solutions to the Schrödinger equation, so is any linear combination of them. For example, if $\psi_1$ is a the solution corresponding to a 100% chance of finding the electron spin up and $\psi_2$ is the solution for a 100% of chance of finding the electron spin down, then $\frac{\sqrt{2}}{2}(\psi_1+\psi_2)$ is also a perfect good solution of Schrödinger's equation; in fact it's the solution in which we have a 50/50 chance of finding the spin up or down. (Don't be misled by the way that the third solution looks more complicated and less "fundamental" then the first too. That's just an accident of the way that I wrote them - all three look like some flavor of $\psi_3\pm\psi_4$ where $\psi_3$ and $\psi_4$ are solutions with 100% probability of finding the spin aligned oin one direction or the other along something other than the vertical or horizontal axes).

It's that probabilistic answer from a single clearly defined and unambiguous wave function that disturbed Einstein (and many other physicists of the era - Einstein was just especially good at articulating the problem). The uncertainty principle comes from a related different source: Those solutions for which the momentum is definite can only be written as a sum of solutions in which the position is definite, and vice versa. Thus, if we set the system up so that there is no uncertainty in the momentum more than one position value has to be possible. The uncertainty principle tells us what the spread in possible position values must be.

For a given energy level, why is it only probable the electron is at the Bohr radius? I thought perhaps the Uncertainty Principle, but I'm wrong.

It's the same thing as above. The states in which the energy is definite are sums of states in which the position has different values, so there some randomness in what number will actually come out of a position measurement. You could say that this is caused by the uncertainty principle, but it's better to think of it and the uncertainty principle as both being caused by the way that solutions to the Schrödinger equation are always sums of other solutions.

 

[a dull boy]

Yes! Thanks so much, I get it!

Mark

 

[a dull boy]

Please allow me one more follow-up - trying to discern superposition from the uncertainty principle. I understand I can't measure position and momentum simultaneously because they are conjugate variables. But what about spin and position, or spin and momentum? I bet you can't simultaneously measure those on a single particle with accuracy, yet I'm not sure why - they should be unrelated.

Mark

 

[bhobba]

Please allow me one more follow-up - trying to discern superposition from the uncertainty principle. I understand I can't measure position and momentum simultaneously because they are conjugate variables. But what about spin and position, or spin and momentum? I bet you can't simultaneously measure those on a single particle with accuracy, yet I'm not sure why - they should be unrelated.

The uncertainty relations are actually a theorem about non-commuting observables:
http://physics.stackexchange.com/questions/10362/how-does-non-commutativity-lead-to-uncertainty

If they commute then there is no uncertainty issue. Spin and position in the direction of the spin commute - momentum and position in the direction of the momentum do not.

As an aside why that is, is quite interesting:
http://physicspages.com/2013/01/15/angular-momentum-commutators-with-position-and-momentum/

Thanks
Bill