Good quantum number
In quantum mechanics, given a particular Hamiltonian and an operator with corresponding eigenvalues and eigenvectors given by , then the numbers (or the eigenvalues) are said to be "good quantum numbers" if every eigenvector remains an eigenvector of with the same eigenvalue as time evolves.
then we require
for all eigenvectors in order to call a good quantum number (where s represent the eigenvectors of the Hamiltonian)
Theorem: A necessary and sufficient condition for q (which is an eigenvalue of an operator O) to be good is that commutes with the Hamiltonian
Proof: Assume .
- If is an eigenvector of , then we have (by definition) that , and so :
- 1 Ehrenfest Theorem and Good Quantum Numbers
- 2 Conservation in Quantum Mechanics
- 3 Analogy with Classical Mechanics
- 4 Systems which can be labelled by good quantum numbers
- 5 Examples
- 6 See also
- 7 References
Ehrenfest Theorem and Good Quantum Numbers
Commonly occurring operators don't depend explicitly on time. If such operators commute with the Hamiltonian, then their expectation value remains constant with time. Now, if the system is in one of the common eigenstates of the operator (and too), then system remains in this eigenstate as time progresses. Any measurement of the quantity will give us the eigenvalue (or the good quantum number) associated with the eigenstates in which the particle is. This is actually a statement of conservation in Quantum Mechanics.
In non-relativistic treatment,and are good quantum numbers but in relativistic quantum mechanics they are no longer good quantum numbers as and do not commute with (in Dirac theory). is a good quantum number in relativistic quantum mechanics as commutes with .
Conservation in Quantum Mechanics
Case I: Stronger statement of conservation : When the system is in one of the common eigenstates of H and A
Let A be an operator which commutes with Hamiltonian. This implies that we can have an orthonormal basis of common eigenvectors of A and H in the vector space consisting of the states of our system. A measurement of A upon the system is bound to yield one of the eigenvalues of A. Assume that our system is in one of the common basis eigenstates of A and H. If we make a measurement of A upon the system, it will definitely yield the eigenvalue of A (or the good quantum number) corresponding to the eigenstate in which the system is. This is because the probability of getting an eigenvalue of an operator is the square of the coefficient attached with the corresponding eigenstate (which in this case is 1) that occurs in the expansion of the state of the system as a linear combination of the basis states. Even if the system is left to evolve for a certain time before the measurement is made, it will still yield the same eigenvalue. This is because the state of the system doesn't change physically (although it does change mathematically; a phase term gets attached). Such states are called stationary states. Now,after one measurement, if we go on measuring A again and again, on the same system, we keep on getting the same value; i.e the same eigenvalue or the good quantum number. This indicates a kind of conservation in Quantum Mechanics. Here the standard deviation of A is zero. A short proof is as follows:
(a is the eigenvalue of A for the state in which the system is. Now, because every measurement of A always yields a )
(Use has been made of the fact that both A and a are hermitian; thus (A-a) is also hermitian; so we can transfer one (A-a) term to the 'bra' side in the inner product.)
Conclusions: The state doesn't change. So, almost every physical quantity is conserved (doesn't change with time)
Case II: Weaker statement of conservation : When the system is not in any of the common eigenstates of H and A
As assumed in case I, [A,H]=0. Also assume that the system is not in any of the common eigenstates of H and A. In this case, the system must be some linear combination of the basis (common) eigenstates of H and A. When a measurement of A is made, it can yield any of the eigenvalues of A. And then, if any number of subsequent measurements of A are made, it is bound to yield the value which was obtained on the first measurement of A. In this case, there holds a (weaker) statement of conservation. Using the Ehrenfest Theorem :
since the operator A commutes with H and we have assumed that A doesn't depend on time explicitly . It says that the expectation value of the operator A remains constant in time. When the measurement is made on identical systems again and again, in general, it will yield different eigenvalues, but for any such identical system the expectation value of A remains constant. It is this kind of conservation that is valid in this case. This is a weaker conservation condition than the case when our system was a common eigenstate of A and H (discussed above).
Conclusions: The system evolves in time. Here fewer quantities are conserved as compared to case I. The quantities that are conserved (expectation value of the operator A) don't have the value equal to the good quantum numbers.
Analogy with Classical Mechanics
This bears striking resemblance to the Ehrenfest Theorem. It implies that a physical quantity is conserved if its Poisson Bracket with the Hamiltonian is zero and it does not depend on time explicitly. This condition in classical mechanics is very similar to the condition in quantum mechanics for the conservation of an observable (as implied by Ehrenfest Theorem:Poisson bracket is replaced by commutator)
Systems which can be labelled by good quantum numbers
Systems which can be labelled by good quantum numbers are actually eigenstates of the Hamiltonian. They are also called stationary states. They are so called because the system remains in the same state as time elapses, in every observable way. The states changes mathematically, since the complex phase factor attached to it changes continuously with time, but it can't be observed.
Such a state satisfies:
- is a quantum state, which is a stationary state;
- is the Hamiltonian operator;
- is the energy eigenvalue of the state .
The evolution of the state ket is governed by the Schrodinger Equation:
It gives the time evolution of the state of the system as:
The hydrogen atom: no spin-orbit coupling
In the case of the hydrogen atom (with the assumption that there is no spin-orbit coupling), the observables that commute with Hamiltonian are the orbital angular momentum, spin angular momentum, the sum of the spin angular momentum and orbital angular momentum, and the components of the above angular momenta. Thus, the good quantum numbers in this case, (which are the eigenvalues of these observables) are . We have omitted , since it always is constant for an electron and carries no significance as far the labeling of states is concerned.
Good quantum numbers and CSCO
However, all the good quantum numbers in the above case of the hydrogen atom (with negligible spin-orbit coupling), namely can't be used simultaneously to specify a state. Here is when CSCO (Complete set of commuting observables) comes into play. Here are some general results which are of general validity :
2. If the observables commute, but don't form a CSCO, then their good quantum numbers refer to a set of states. In this case they don't refer to a state uniquely.
3. If the observables don't commute they can't even be used to refer to any set of states, let alone refer to any unique state.
In the case of hygrogen atom, the don't form a commuting set. But are the quantum numbers of a CSCO. So, are in this case, they form a set of good quantum numbers. Similarly, too form a set of good quantum numbers.
The hydrogen atom: spin-orbit interaction included
Now, the new Hamiltonian with this new term doesn't commute with and ; but it does commute with L2, S2 and , which is the total angular momentum. In other words, are no longer good quantum numbers, but are.
As we can see, the above expressions contain the good quantum numbers, namely
- Complete set of commuting observables
- Hamiltonian (quantum mechanics)
- Stationary state
- Constant of motion
- Quantum number
- Measurement in quantum mechanics
- Ehrenfest theorem
- Operator (physics)
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- Poole, Herbert Goldstein, Charles P. (2001). Classical mechanics, 3e (3rd. ed. ed.). United States: PEARSON EDUC (HIGHER ED GRP)(BOX 70632) (NJ). p. 396. ISBN 0201657023.
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