Confusion with Wikipedia text

[omarshehab]

I think my knowledge in QM is even less than elementary. For example, right now I am reading the following article.

http://en.wikipedia.org/wiki/Quantum_mechanics

Here are my confusions about the text:

1. Section 'Quantum mechanics and classical physics', second paragraph: The line is "Essentially the difference boils down to the statement that quantum mechanics is coherent (addition of amplitudes), whereas classical theories are incoherent (addition of intensities).". I can't visualize what does it physically mean.
2. Section 'Theory', third paragraph: The line is "However, quantum mechanics does not pinpoint the exact values of a particle for its position and momentum (since they are conjugate pairs) or its energy and time (since they too are conjugate pairs); rather, it only provides a range of probabilities of where that particle might be given its momentum and momentum probability." I am clear with the conjugate pair of position and momentum. But how time and energy are also conjugate?

 

[tiny-tim]

Welcome to PF!

Hi omarshehab! Welcome to PF!

1. Section 'http://en.wikipedia.org/wiki/Quantum_mechanics#Quantum_mechanics_and_classical_physics"', second paragraph: The line is "Essentially the difference boils down to the statement that quantum mechanics is coherent (addition of amplitudes), whereas classical theories are incoherent (addition of intensities).". I can't visualize what does it physically mean.

Yes, it's very unclear.

(And it doesn't help that it follows on from "The main differences between classical and quantum theories have already been mentioned above in the remarks on the Einstein-Podolsky-Rosen paradox." when that isn't "above"! )

It physically means that in classical physics, probablitites add, but in quantum physics they don't (in fact they sometimes cancel) … the well-known experimental result which shows this is the double-slit experiment.

This wikipedia page is unclear … I strongly suggest you start with http://en.wikipedia.org/wiki/Double_slit instead.

… I am clear with the conjugate pair of position and momentum. But how time and energy are also conjugate?

Do you understand 4-vectors? If not, have a look at http://en.wikipedia.org/wiki/4-momentum and http://en.wikipedia.org/wiki/Four-vector

 

[D H]

Regarding conjugate pairs: There are lots of ways to represent the coordinates of something in addition to good old x, y, z positional coordinates. For example, angles are often a good choice when dealing with rotational behavior. An appropriate choice of coordinates can make solving a problem easy.

The concept of generalized coordinates is critical for Lagrange's and Hamilton's formulations of physics. Lagrangian dynamics is written in terms of the Lagrangian $L$, the difference between the total kinetic and potential energy for the system; some generalized coordinates $q$; and its time derivative $\dot q$, called generalized velocity.

The Hamiltonian formulation of physics extends Lagrangian dynamics. It starts with the Lagrangian $L$ and some set of generalized coordinates $q$. Hamilton's formulation introduces the concept of generalized momentum $p$. Each element of the generalized momentum vector is the partial derivative of the Lagrangian with respect to an element of the generalized velocity:

$$p_j = \frac {\partial L}{\partial \dot q_j}$$

The generalized coordinates and generalized momentum form a conjugate pair. Since the Lagrangian L has units of energy, the product of generalized velocity and generalized momentum will also have units of energy. This means the product of generalized coordinates and generalized momentum will have units of energy*time. Any pair of variables that has units of energy*time is a candidate for forming a conjugate pair. (This is a necessary but not sufficient condition.)