What is the meaning of quantum in quantum mechanics?

[c299792458]

There are many $$n$$'s in QM, and I am confused as to which is which. For example, there are:

• $$n$$ called principal quantum number
• $$n_r=n-l-1$$ (anonymous n)
• $$n_i$$ in $$E=({3\over2}+n_x+n_y+n_z)$$ and $$n=\sum_i n_i$$
• $$n$$ the indices of energy as in $$E_n$$
• and lots more

Could anyone kindly help me distinguish them? Some of them take on values including 0 while others don't. Very confusing!

Thanks

 

[jtbell]

What the integer n means, depends on the equation and what it's used for.

 

[c299792458]

How is the principal quantum number related to N in $$E=\hbar\omega({3\over2}+N)$$? (If there is indeed a relation)

 

[kith]

Mostly, n is used to number discrete eigenvalues of the Hamiltonian.

One important Hamiltonian is that of the hydrogen arom. Here, the term 'principle quantum number' is used and so is your n_r, which is related with the ordinary n via your formula.

Another important Hamiltonian is that of the harmonic oscillator. You have written out the eigenenergies for the 3D case, so you have one label n_i for every spatial direction. The n here is not related to the principle quantum number, which belongs to the hydrogen atom (or electronic states in atoms in general) and therefore to a different physical system.

 

[Ken G]

I think this question can be framed another way: just what is the "quantum" in quantum mechanics anyway? In fact it is quite a few different things, and this does indeed get quite confusing. Sometimes the "quantum" is in "quantum number", which has to do with the discrete spectrum of possible outcomes when you do a measurement. But other times, like in the two-slit experiment, it has to do with the fact that a particle is a "quantum" that has to show up at only one place. Will the real "quantum" please stand up?