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Aerozeppelin
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Where l is the azimuthal quantum number!
Nugatory said:the total magnitude of that vector is ##l##.
Nugatory said:Intuitively, the ##m## quantum number is the value of the ##z## component of the angular momentum vector, and the total magnitude of that vector is ##l##. The magnitude of a single component of a vector has to be less than the magnitude of the vector;
I totally understand it now, great explanation!Nugatory said:Intuitively, the ##m## quantum number is the value of the ##z## component of the angular momentum vector, and the total magnitude of that vector is ##l##. The magnitude of a single component of a vector has to be less than the magnitude of the vector; if I am moving in some direction at a speed of 10 m/sec, we don't know what my speed in a northwards direction is, but we know it has to be somewhere between -10 m/sec (I'm moving due south) and 10 m/sec (I'm moving due north).
Mathematically, that's what falls out of the solution of the Schrodinger equation for the electron in the electric field of the nucleus. Google for "hydrogen atom Schodinger" to see how this works.
Aerozeppelin said:I totally understand it now, great explanation!
jtbell said:More precisely (although it doesn't change the conclusion), the magnitude and z-component of the angular momentum are
$$L = \sqrt{l(l+1)} \hbar\\
L_z = m \hbar$$
It's easy to see that if m were to equal ##l+1##, then we would have ##L_z > L##.
Aerozeppelin said:Lz = m [itex]\hbar[/itex]
This means that the angular momentum is quantized right?
Would a magnetic/electric field alter this relationship?
Aerozeppelin said:Would a magnetic/electric field alter this relationship?
jtbell said:It means that one component of the angular momentum is quantized. By convention we usually call this the "z-component". The other two components (x and y) are indeterminate.
WannabeNewton said:No that is an inherent property of the angular momentum operator i.e. it is entirely independent of what system you consider. It is the Hamiltonian which depends on the system and accordingly it is the energy eigenvalues that will be shifted in the presence of electric or magnetic fields (c.f. Stark effect, Zeeman effect, fine structure of Hydrogen).
Aerozeppelin said:Ok, I'm finding the whole spatial quantisation concept pretty difficult to grasp.
oops, I meant eigenvalue of the angular momentum squared operator. So ##\hat{L^2}[fg]=\hbar \sqrt{l(l+)} [fg]##. Excuse the brain fart, I just started cramming introductory Quantum physics for the exams like a few days ago.Nikitin said:On the rhs, ##g(\phi)## will disappear after some algebra, and you'll end up with legendre differential equation http://en.wikipedia.org/wiki/Legendre_polynomials for ##f(\theta)##. From that you'll find that the eigenvalue of ##f(\theta)## must be quantized.
The magnetic quantum number is related to the orientation of an electron's spin in a magnetic field. The value of the magnetic quantum number is limited to between -l and +l because this range represents all possible orientations of an electron's spin in a magnetic field. Any value outside of this range would result in an invalid orientation.
The value of the magnetic quantum number is determined by the principal quantum number, l, which represents the energy level of the electron. The magnetic quantum number can have any value between -l and +l, including 0, and it helps to differentiate between the different sublevels within an energy level.
The magnetic quantum number does not directly affect an electron's behavior, but it does provide information about the electron's orientation and energy level. This information can be used to understand an electron's interactions with a magnetic field and its potential energy within an atom.
Yes, the magnetic quantum number can change as an electron absorbs or emits energy and moves between different energy levels. However, the value of the magnetic quantum number will always fall within the range of -l to +l.
The magnetic quantum number is important in quantum mechanics because it helps to describe the energy levels and sublevels of electrons within an atom. This information is essential in understanding the behavior and interactions of electrons, which ultimately determine the properties and behavior of matter.