# QM probability and normal distribution

1. Oct 4, 2015

### forcefield

Is there a relationship between QM probability and normal distribution ?

I'm thinking about drawing probability densities as functions of phase.

Thanks

2. Oct 4, 2015

### Staff: Mentor

You can find things that follow a normal distribution in QM, but I would not call this "relationship between QM and normal distribution" in the same way I don't see a "relationship between the normal distribution and the number 3".

3. Oct 4, 2015

### forcefield

You dropped out an essential word which was "probability". I'm looking for a mathematical relationship to calculate QM probabilities from probability densities.

4. Oct 4, 2015

### Staff: Mentor

Probability densitites are used in quantum mechanics, but quantum mechanics is more than pure probability theory.

5. Oct 4, 2015

### forcefield

That's why I said QM probability.

6. Oct 4, 2015

### Simon Phoenix

Hi Forcefield, I'm really not at all sure what you're trying to get at here.

The probability function you get depends on what observable you're trying to measure - and the state of the system. Suppose we prepared a single qubit (a spin-1/2 particle say) in an eigenstate of the spin-z operator, and let's suppose we prepared the spin up state. Then, assuming ideal measurements, the probability of getting spin up in a measurement of spin-z is 1, and the probability of getting spin down is 0. Not really very Gaussian :-)

Now suppose we measure spin-x, then the probability of getting a spin up result is now 1/2 and the probability of getting a spin down result is 1/2. A uniformly random distribution.

Let's take some more examples - if we have a coherent state of light and make a measurement of photon number - then we'll get a Poisson distribution (in many runs of the same experiments, of course). Measurement of the field quadrature operator of the same coherent state will give us a Gaussian (if I recall correctly). Take a photon number state of the EM field and measure it's phase - you'll get a uniformly random distribution.

So the probabilities depend crucially on what property we're choosing to measure and what state the system is in.

Could you perhaps be a bit more specific because your question doesn't make a lot of sense to me.

7. Oct 4, 2015

### forcefield

Hi Simon,

Actually I think it is Gaussian but the variance is zero.

I think that makes it Gaussian too just like throwing coins. It is just difficult to see because there are only two options.

I have only a vague idea what you are talking about here. Remember I marked this thread high school level.

You know there is this so called phase that you can throw away when you calculate probabilities in QM. I am thinking about whether it is possible to calculate QM probabilities by assuming that the measurement probability changes like normal distribution as time goes by.

8. Oct 4, 2015

### blue_leaf77

Given $\psi(x,t)$, you want $|\psi(x,t)|^2$ at fixed position $x=x_0$ to follow Gaussian profile with time, is that what you mean?

9. Oct 4, 2015

### Simon Phoenix

You're talking here about a global phase factor - you certainly can't ignore the relative phases between terms in a superposition when calculating probabilities. Recall that to calculate a probability in QM we're taking the square modulus of sums of complex numbers.

Well only in the sense that for sufficiently large numbers of trials then the Gaussian is a good approximation to the binomial (I think the rule of thumb is that np ≥ 5 where n is the number of trials and p is the probability)

10. Oct 4, 2015

### Staff: Mentor

Only phases in the basis where you want to calculate the probabilities, at the time when you want to calculate them.
No.

11. Oct 4, 2015

### forcefield

No, I'm not considering wave functions. Actually I haven't studied them seriously yet.

12. Oct 4, 2015

### forcefield

Yes, I thought about that and it looks like a relative phase of pi/2 is required.

13. Oct 4, 2015

### Staff: Mentor

The relative phase in a superposition can be everything (well, as it is a phase, it is usually restricted to the range from 0 to 2pi).

14. Oct 4, 2015

### blue_leaf77

The only entity which connects QM with probabilistic interpretation is the wavefunction.

15. Oct 4, 2015

### forcefield

I think one should consider a period of π here.

16. Oct 4, 2015

### Staff: Mentor

That does not make sense.

17. Oct 5, 2015

### forcefield

Yes, I was sloppy there. The length of the period doesn't really matter but if you look at the shape of normal distribution it is more like half-circle than full-circle.

18. Oct 5, 2015

### Heinera

Sorry, but that made even less sense.