Thread: Uncertainty Principle

So you often see this explained by the apparent fact that as soon as you make a measurement on an elementary particle then you of necessity alter its state. This is fine as far as it goes, but it is as well to remember this is a mere analogy (I believe it was suggested to Heisenberg by Pauli, though I am not completely sure)

However there are some problems here: ask a modern physicist and she will probably say something like "a quantum entity is neither a wave nor a particle but, depending on how you measure it could show some properties of either one or the other or both simultaneously"

Let us take a more simplistic view, one that both de Broglie and Schroedinger took.: to every elementary particle one can associate a wave function .

Now it is a fact that the playground of Quantum Mechanics is an Hilbert space of square-integrable functions called "state vectors", so let us assume that our wave function is one such. To celebrate, let's write this as . As this is a vector, it is entitled to be acted upon by some operator, so let us, following Schroedinger choose the Hamiltonian with the property that

Never mind what actually is, just note that the is called an "energy eigenvalue" associated to this operator.

Let us now consider what our wave function/state vector is. It is a function that takes on values in with period where n is integer.

OK, so forget the "eigen" bit, and assume that to any particle, via its associated wave functions, there may be many possible energies, and a prior we have no way of knowing which wave function/energy to assign to it. But the process of measuring energy is just to select a single wave function.

Here's the crucial point: it was first pointed out by Max Born (I think) that one may interpret the square of the wave function as a position probability density for our particle. So that having measured its energy, and thereby selected a PARTICULAR wave, all we can say about its position is in terms of this probability density.

Conversely, assume we know exactly where our particle is. Then our probability density will be unity at some point, and zero everywhere else. The only way to obtain such a wave with the properties I describe is to "muddle together" (physicists call this superposition) a large number of different waves that interfere destructively at all points except one, where interference is constructive and square all of these.

In other words, if you know the position, you are faced with a multiplicity of possible energies, and no way to decide between them

I think this Uncertainty is like Niels Bohr doig his over simplistic model of the atoms, just that it would be hard to calculate the exact behaviour of super strings, thus we might have to wait a few thousands years to study atoms on that low level.

Well, I confess I understood very little of that HexHammer. Never mind.....

Recall I said I would not bother you with the details of Schroedinger's Hamiltonian, but it occurs to me there is a more accurate, but more abstract and therefore less intuitive way of stating the uncertainty principle.

So the operator is by definition the sum of a kinetic term and a potential term, where, in its most primitive form, we can take these "terms" to be the respective energies.

Now it is a fact from elementary operator theory that whenever an operator can be decomposed as the sum of parts, then these parts must themselves be operators.

It is obvious there is a connection between momentum and kinetic energy. Like , so let's write this "sub-operator" as

Equally obvious is that potential energy in some way depends upon position. So let's define the position operator as

So the Uncertainty Principle can be expressed by , or by saying that the momentum and position operators do not commute.

Which is just a fancy way of saying that changing the order in which momentum and position are measured will give a different result..

What I tryed to say in my former post, is that I don't belive in randomness not even for superstrings, therefore when we gain understanding on the excat behaviour of superstrings, we can eliminate this uncertainty principle.


[note to self] dont' write anything when drunk.

why exactly do we need hamiltonian operators again? Why not simply write the equation to describe the physical situation as you wish to express it?

Originally Posted by HexHammer

What I tryed to say in my former post, is that I don't belive in randomness not even for superstrings, therefore when we gain understanding on the excat behaviour of superstrings, we can eliminate this uncertainty principle.

Your beliefs are irrelevant to what is actually possible.

Originally Posted by ballyhoo

why exactly do we need hamiltonian operators again? Why not simply write the equation to describe the physical situation as you wish to express it?

Hamiltonian operators do provide an equation that describes a physical situation and can be solved to determine the energies associated with possible wavefunctions of the system. See Schrodinger equation.

usually though, in QM are they used to transform another equation? https://en.wikipedia.org/wiki/Operator_%28physics%29

I'm wondering why bother at all to apply a hamiltonian and when do you know when to apply it?

Lets see if I can get this right, the Hamiltonian is used in determining the wave function or the energy eigenvalues associated with a quantum system. The most common example of this is the particle in a box, which may be used to find the energy levels of an electron in each s-orbital of hydrogen, however there are many other examples that involve quantum systems that are subject to some form of potential. (magnetic dipole interactions, energy levels in many electron atoms etc.)

So extending on the electron energy level example, the eigenvalues of the Hamiltonian describe the possible energy levels of an electron in the atom. Prior to observation the electron could be in any of the associated eigenstates, after we observe which energy level it is in however the wave function collapses to just one eigenstate.

Hopefully i'm not muddying the waters with this attempted explanation, it's a bit of a side issue to the uncertainty principle either way.

This is good Wallers, but for a small quibble

Originally Posted by wallaby

the Hamiltonian is used in determining the wave function or the energy eigenvalues associated with a quantum system.

The Hamiltonian does not "determine" the wave function, but you are correct; it does determine the allowable energies associated to a quantized system.

In fact, it is an axiom of QM that, for any measurable property of a quantum system, any measurement of this property will be an eigenvalue for some Hermitian operator acting on an Hilbert space of state vectors.

Recall an Hermitian operator is one that is equal to its own conjugate transpose

Originally Posted by Guitarist

The Hamiltonian does not "determine" the wave function, but you are correct; it does determine the allowable energies associated to a quantized system.

True, i think i should have stated that the Hamiltonian is a key part of the Schrödinger equation. (The solutions of said equation being the eigenfunctions associated with the eigenvalues of the Hamiltonian)

I wonder if anyone here is familiar with Commercial Quantum Theory ?

This states that a set of corporate accounts is deemed completely Kosher right up to the instant
they are scrutinised at which point, the profit / loss duality is reduced to a finite quantity.

The phenomenon is known as 'Corporate share collpase' : )

Lets try and stick to physics here, OK?