Examples of quantum randomness?

[houlahound]

hope I am not totally lost here but can someone give specific examples of randomness re quantum mechanics.

in my view any well constructed identical experiments that are repeated 1000 times yield the same well defined very sharp distribution of results within the precision of the experimental set up, at least to my knowledge.

the ionisation energy of hydrogen has a quoted value, far from random.

apologies if this is off topic.

 

[Nugatory]

in my view any well constructed identical experiments that are repeated 1000 times yield the same well defined very sharp distribution of results within the precision of the experimental set up, at least to my knowledge.

Consider an experiment in which a spin-up particle is fired at a detector set to measure spin in the horizontal direction. If we perform this measurement many thousands of times we find that the result is spin-left half the time and spin-right half the the time. This is the essential randomness of quantum mechanics: the spin-up state is one for which the Born rule predicts that anyone measurement of the horizontal spin will be randomly spin-left or spin-right.

 

[Jilang]

The time it takes for an unstable particle to decay would be another example.

 

[jack476]

This was a thought experiment invented by Richard Feynman in the Feynman Lectures.

Set up a double slit experiment kit and fire a single electron through the double slit. Record where the electron hits the screen behind the slit. Reset the experiment and fire another single electron. It will strike the screen in a different location. After a large number of trials, you will see that the pattern made by where the electrons strike the screen resembles the diffraction pattern that would result from classical interference of waves (for instance, if you had used a continuous stream of electrons).

This occurs because the individual electron does not have a classical trajectory (or else it would strike the screen in the same place each time), but rather it has a wave function which returns a probability amplitude at each location, and it is this wave function that is subject to interference.

Another really cool example comes from this entry in the Harvard physics problems. I'll just post the problem itself, but it requires some fairly difficult math so you can just read the solution if you just want the result, though I strongly recommend trying the problem yourself if you can.

Basic problem statement: Consider a pencil balanced perfectly upright on a flat surface. Taking Heisenberg's Uncertainty Principle into consideration, for roughly (to a first-order approximation) how long is it possible for the pencil to stand straight up?

https://www.physics.harvard.edu/uploads/files/undergrad/probweek/prob3.pdf

Spoiler

Three and a half seconds.
https://www.physics.harvard.edu/uploads/files/undergrad/probweek/sol3.pdf

 

[houlahound]

All of those examples have results that fit inside well defined distributions that can be calculated in advance and it is not a random distribution.

There must be a special use of the word random being employed.

 

[DrChinese]

All of those examples have results that fit inside well defined distributions that can be calculated in advance and it is not a random distribution.

They are random distributions, even if the distribution itself fits a Bell curve or whatever.

In your OP, you mentioned a sample of 1000. Not likely you would see the same 1000 results twice in a row, for example.

 

[Nugatory]

There must be a special use of the word random being employed.

It's the Born rule: Given a system prepared in the completely specified initial state $|\psi\rangle=c_1|\psi_1\rangle+c_2|\psi_2\rangle+...c_n|\psi_n\rangle$ where the $|\psi_i\rangle$ are the eigenfunctions with eigenvalue $a_i$ of an observable $A$, a measurement of $A$ will randomly yield one of the $a_i$ with probability $|c_i|^2$.

This is in stark contrast with classical mechanics. There, given a system prepared in some completely specified initial state, a measurement can only yield a single result and it will yield that result every time with the determinism and predictability of well-engineered clockwork. Any apparent randomness (as with, for example, a tossed coin) just means that we've failed to completely specify the initial state; the apparent randomness is the result of something that we missed in setting up the experiment.

 

[houlahound]

my bold

They are random distributions, even if the distribution itself fits a Bell curve or whatever.

to me that is contradictory ie A = - A

 

[houlahound]

It's the Born rule: Given a system prepared in the completely specified initial state $|\psi\rangle=c_1|\psi_1\rangle+c_2|\psi_2\rangle+...c_n|\psi_n\rangle$ where the $|\psi_i\rangle$ are the eigenfunctions with eigenvalue $a_i$ of an observable $A$, a measurement of $A$ will randomly yield one of the $a_i$ with probability $|c_i|^2$.

i won't argue further on this because i think there must be different semantics happening, the above quote, in my mind, negates the system as random, in fact it proves it is not random.

 

[houlahound]

the following is a random distribution in my use of the word, the Ci in the quoted equation above will clearly have a spectra with more structure than the random spectra below.

 

[jack476]

Okay, here's one that's truly random. Australian National University's physics department hosts an online service that uses vacuum fluctuations to provide the seed for a true random number generator: https://qrng.anu.edu.au/

 

[Nugatory]

i won't argue further on this because i think there must be different semantics happening, the above quote, in my mind, negates the system as random, in fact it proves it is not random.

Let's consider a variant of Schrödinger's cat:

We set up several thousand of Schrödinger's boxes, each containing a sample of radioactive material that has a 50% probability of decaying in the next ten minutes; if it does, it releases the cyanide and kills whoever is in the box. We are going to lock you up for ten minutes in one of those boxes. This is bad news for you - it looks as if there is a 50% chance that you are going to die in one of these boxes.

But we'll give you a chance to save yourself! You get to choose which box, and we will give you arbitrarily precise measurements of the exact state of every single subatomic particle in the radioactive sample in each box, and we will give you computers as powerful as you want and unlimited time to make calculations based on this information before you choose.

In classical physics, you can save yourself: Just crunch the data for each box until you find one of the five hundred or so in which the laws of physics say the radioactive sample won't decay in the next ten minutes, and choose that box. There's no randomness if you understand the problem deeply enough - some boxes will kill their occupants, some won't, and you can calculate which are which up front.

Quantum mechanics says that no matter how much data about the initial state you have, and no matter how much time you spend crunching it, and no matter how much advanced physics you apply to the problem, you'll get the same answer for every box: 50% chance that the decay will happen in the next ten minutes. You can't make your chances of survival better than a coin toss.

This difference is what people are referring to when they say that there is a unavoidable randomness in quantum mechanics. You may want to want use another word for that difference - but there is a difference between getting into the box when you know you'll live and when you know there's a 50% chance that you won't live.

 

[houlahound]

thanks for reply, I think I am getting your point. however I still feel that a 50:50 is not random in the way I am know the word eg;

a 50:50 chance of rain is vastly different to all probabilities of getting rain from 0 - 100% are equally likely which is what random suggests.

as for the decay in your example I think you have built in a bias eg the decay rate is something like

N(t)=Ne^(-kt)

k is not a random number, but yeah agree which nucleus will decay next is unpredictable.

my feeling is there is equivocation on the word unpredictable and the word random.

one is a knowledge claim, the other is an equal distribution of outcomes.

 

[DrChinese]

a 50:50 chance of rain is vastly different to all probabilities of getting rain from 0 - 100% are equally likely which is what random suggests.

You have selected a definition which makes your statements true to you.

Random may be defined a number of ways, and Nugatory has already provided a nice explanation as it relates to physics. If you reject the examples you requested in the OP - all of which you have been told are examples of random particle behavior - then how can we assist?

In the context of quantum physics: many measurements of particle attributes yield values which have no known prior cause. Preparing 2 particles in identical states will not yield identical outcomes. The selection of one value vs. another is purely random.

 

[houlahound]

understand now, it was a difference in semantics on the word random. I was using the word in a different, more general, way.

cheers