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2007-05-22 22:50:54 · 3 answers · asked by sks 1 in Science & Mathematics ➔ Chemistry
There are a set of quantum numbers associated with the energy states of the atom. The four quantum numbers n, l, m, and s specify the complete and unique quantum state of a single electron in an atom called its wavefunction or orbital. The wavefunction of the Schrödinger wave equation reduces to the three equations that when solved lead to the first three quantum numbers. Therefore, the equations for the first three quantum numbers are all interrelated. The magnetic quantum number arose in the solution of the azimuthal part of the wave equation as shown below.
The magnetic quantum number associated with the quantum state is designated as m. The quantum number m refers, loosely, to the direction of the angular momentum vector. The magnetic quantum number m does not affect the electron's energy, but it does affect the probability cloud. Given a particular , m is entitled to be any integer from up to . More precisely, for a given orbital momentum quantum number (representing the azimuthal quantum number associated with angular momentum), there are integral magnetic quantum numbers m ranging from to , which restrict the fraction of the total angular momentum along the quantization axis so that they are limited to the values m. This phenomenon is known as space quantization. It was first demonstrated by two German physicists, Otto Stern and Walther Gerlach.
To describe the magnetic quantum number m you begin with an atomic electron's angular momentum, L, which is related to its quantum number by the following equation:
L= h' ( l*(l+1) )^1/2
where h'=h/2 pie is Planck's reduced constant, also called Dirac's constant. The energy of any wave is the frequency multiplied by Planck's constant. This causes the wave to display particle-like packets of energy called quanta. To show each of the quantum numbers in the quantum state, the formulae for each quantum number include Planck's reduced constant which only allows particular or discrete or quantized energy levels.
To show that only certain discrete amounts of angular momentum are allowed, has to be an integer. The quantum number m refers to the projection of the angular momentum for any given direction, conventionally called the z direction. Lz, the component of angular momentum in the z direction, is given by the formula:
Lz = m h'
Another way of stating the formula for the magnetic quantum number (ml = , …, 0, …, ) is the eigenvalue, Jz=mlh/2π.
Where the quantum number is the subshell, the magnetic number m represents the number of possible values for available energy levels of that subshell as shown in the table below.
For d orbitals the magnetic quantum numbers are -2,-1,0,+1, +2;
for 3d (x^2-y^2 ) it would be +1.
2007-05-30 22:51:53 · answer #1 · answered by sb 7 · 0⤊ 0⤋
I'm sorry to tell you this, but the 3d(x^2-y^2) orbital is used as part of the REAL representation of the spherical harmonics. That means that it is a superposition of m=2 and m=-2. Are you sure you're not talking about L? In that case the answer is most definitely 2 for all d orbitals.
2007-05-23 08:09:12 · answer #2 · answered by supastremph 6 · 0⤊ 0⤋
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2007-05-23 05:58:31 · answer #3 · answered by law_student_to_be 1 · 0⤊ 0⤋