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Quantum number
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Quantum numbers describe values of conserved numbers in the dynamics of the quantum system. They often describe specifically the energies of electrons in atoms, but other possibilities include angular momentum, spin etc. Since any quantum system can have one or more quantum numbers, it is a futile job to list all possible quantum numbers.
How Many Quantum Numbers? The question of how many quantum numbers are needed to describe any given system has no universal answer, although for each system one must find the answer for a full analysis of the system. The dynamics of any quantum system are described by a quantum Hamiltonian, H. There is one quantum number of the system corresponding to the energy, i.e., the eigenvalue of the Hamiltonian. There is also one quantum number for each operator O that commutes with the Hamiltonian (i.e. satisfies the relation OH = HO). These are all the quantum numbers that the system can have. Note that the operators O defining the quantum numbers should be independent of each other. Often there is more than one way to choose a set of independent operators. Consequently, in different situations different sets of quantum numbers may be used for the description of the same system.
Quantum numbers with spin-orbit interactionWhen one takes the spin-orbit interaction into consideration, l, m and s no longer commute with the Hamiltonian, and their value therefore changes over time. Thus another set of quantum numbers should be used. This set includes
- The (j = 1/2,3/2 ... n−1/2) gives the total angular momentum through the relation .
- The (mj = -j,-j+1... j), which is analogous to m, and satisfies .
- Parity. This is the eigenvalue under reflection, and is positive (i.e. +1) for states which came from even l and negative (i.e. -1) for states which came from odd l. The former is also known as even parity and the latter as odd parity
For example, consider the following eight states, defined by their quantum numbers:
- (1) l = 1, ml = 1, ms = +1/2
- (2) l = 1, ml = 1, ms = -1/2
- (3) l = 1, ml = 0, ms = +1/2
- (4) l = 1, ml = 0, ms = -1/2
- (5) l = 1, ml = -1, ms = +1/2
- (6) l = 1, ml = -1, ms = -1/2
- (7) l = 0, ml = 0, ms = +1/2
- (8) l = 0, ml = 0, ms = -1/2
The quantum states in the system can be described as linear combination of these eight states. However, in the presence of spin-orbit interaction, if one wants to describe the same system by eight states which are eigenvectors of the Hamiltonian (i.e. each represents a state which does not mix with others over time), we should consider the following eight states:
- j = 3/2, mj = 3/2, odd parity (coming from state (1) above)
- j = 3/2, mj = 1/2, odd parity (coming from states (2) and (3) above)
- j = 3/2, mj = -1/2, odd parity (coming from states (4) and (5) above)
- j = 3/2, mj = -3/2, odd parity (coming from state (6))
- j = 1/2, mj = 1/2, odd parity (coming from state (2) and (3) above)
- j = 1/2, mj = -1/2, odd parity (coming from states (4) and (5) above)
- j = 1/2, mj = 1/2, even parity (coming from state (7) above)
- j = 1/2, mj = -1/2, even parity (coming from state (8) above)
General principles
Particle physics
See also
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