Quantum Tunneling and atomic spectra.

[misogynisticfeminist]

Can somebody explain quantum tunneling to me? And the thing about why amplitudes when they are trapped, the energies must choose from a distinct set of values? And why when particles that are totally free to wander will have any energy that they like?

Also, why in the atomic spectra, more than two electrons with opposite spins can fit into the second and subsequent energy levels? How many electrons can fit into each level? And why is this so?

 

[marlon]

Can somebody explain quantum tunneling to me? And the thing about why amplitudes when they are trapped, the energies must choose from a distinct set of values? And why when particles that are totally free to wander will have any energy that they like?

Also, why in the atomic spectra, more than two electrons with opposite spins can fit into the second and subsequent energy levels? How many electrons can fit into each state? And why is this so?

Tunneling is the fenomenon in QM that some particles can get through a potential barrier. In classical mechanics this is impossible because the particles would require negative kinetic energy to do so. This implies negative mass-values.

In the barrier there is a discrete set of possible energyvalues, yes. this is proven by solving the Schrödingerequation in such a barrier. The energy-levels have to be discrete or otherwise the wavefantion of the particle cannot be finite, thus unfysical.

Only two electrons with opposite spin per energylevel !
The reason for this is the Pauli-exclusionprinciple that states that no two particles with the exact same set of quantumnumbers can be found in the same QM-state. This condition is necessary because the wavefunctions of baryons and leptons like electrons (matter) are antisymmetric.


I can give you exact calculations if you wa,t, though they can be found in any standard QM-textbook

regards
marlon

 

[vanesch]

Only two electrons with opposite spin per energylevel !

Except if the energy level is degenerate of course...
I think the OP was well aware of the above and wondered why you can have MORE than 2 electrons per energy level. In the simple coulomb hydrogen atom (no LS couplings or whatever), for energy level n (1, 2, 3...), you can have a value of l from 0 to n-1 and for each value of l you can have a value of m from -l to l. The THREE numbers (n, l , m) specify an ENERGY STATE (without spin). However, only the first number determines the ENERGY LEVEL (the value). When such a thing happens, we talk of degeneracy (many different states correspond to the same value). The Pauli exclusion principle only allows 2 electrons (spin up and spin down) per STATE. But as (except for n = 1) you can have several states for the same value, you can have more than 2 electrons per energy value.

Now, one should add that when you take more effects into account than just the electrostatic potential, such as the magnetic coupling of the electron spin to the E-field (because the electron moves), the spin-spin coupling between nucleus and so on... it turns out that a lot of the degeneracy is lifted: the different energy states which, for the coulomb interaction, had the same energy level start to have slightly different values.

cheers,
Patrick.