How Do Quantum Numbers Shape Atomic Structure?

[cam875]

im just learning the quantum model of the atom now and i have a few questions,

1st: why does the different energy levels represented by the primary quantum number n have different numbers of subshells represented by l such as 0,1,2,3. Energy level 1 can only have s or 0 but Energy Level 2 can have 0 or 1 which is s and p. Why is this, what is the difference between the two energy levels that causes this to be possible.

2nd: I don't understand what the m is for, it stands for magnetic or something but I am just confused about it, could someone explain that part of the 4 quantum numbers for describing an electron.

Thanks in advance.

 

[malawi_glenn]

i) It comes from the math, theory of differential equations, when you solve this Shcrödinger equation.

You can think "semiclassical" about this, n is the radial quantum number, and the larger it is, the larger angular momentum is permitted. (if we think of n as distance from centre). But the real answer is what you obtain when you solve the shcördinger equation.

Have you done class in partial differential equations? if not, maybe wait til then to try to get the solution buy yourself.

ii) m is the projection of l on the z-axis which is, according to QM, quantized.

http://en.wikipedia.org/wiki/Principal_quantum_number
http://en.wikipedia.org/wiki/Azimuthal_quantum_number
http://en.wikipedia.org/wiki/Magnetic_quantum_number

 

[cam875]

no i have not done differential equations yet so yeah ill just wait for that, I am assuming that's where a lot of advanced physics equations are based around?

 

[malawi_glenn]

yes, that is true. Knowing theory of differential equations, orthogonal functions etc. are essential for a physicsist

 

[jtbell]

no i have not done differential equations yet so yeah ill just wait for that, I am assuming that's where a lot of advanced physics equations are based around?

Yes indeed. In mechanics, you solve Newton's Second Law as a differential equation for position as a function of time. In electricity & magnetism, Maxwell's equations are differential equations for the components of the electric and magnetic fields, as functions of position and time. In quantum mechanics, Schrödinger's equation is a differential equation for "psi" as a function of position versus time.

For QM, if you know basic calculus, you might try a second-year university "modern physics" textbook. They often introduce the concepts of differential equations specifically in connection with Schrödinger's equation, and show one or two simple solutions. For example, Beiser's "Concepts of Modern Physics." It has a chapter on the hydrogen atom, which gives the key steps and results from solving the SE for hydrogen, without going into all the mathematical details, which most students probably don't see until graduate school.

 

[cam875]

ill be doing calculus next year so I am not jumping into this just yet :)