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In
quantum mechanics Quantum mechanics is a fundamental theory in physics that provides a description of the physical properties of nature at the scale of atoms and subatomic particles. It is the foundation of all quantum physics including quantum chemistr ...
, given a particular
Hamiltonian Hamiltonian may refer to: * Hamiltonian mechanics, a function that represents the total energy of a system * Hamiltonian (quantum mechanics), an operator corresponding to the total energy of that system ** Dyall Hamiltonian, a modified Hamiltonian ...
H and an operator O with corresponding eigenvalues and
eigenvectors In linear algebra, an eigenvector () or characteristic vector of a linear transformation is a nonzero vector that changes at most by a scalar factor when that linear transformation is applied to it. The corresponding eigenvalue, often denoted ...
given by O, q_j\rangle=q_j, q_j\rangle, the q_j are said to be good quantum numbers if every eigenvector , q_j\rangle remains an eigenvector of O ''with the same eigenvalue'' as time evolves. In other words, the eigenvalues q_j are good quantum numbers if the corresponding operator O is a constant of motion. Good quantum numbers are often used to label initial and final states in experiments. For example, in particle colliders: 1. Particles are initially prepared in approximate momentum eigenstates; the particle momentum being a good quantum number for non-interacting particles. 2. The particles are made to collide. At this point, the momentum of each particle is undergoing change and thus the particles’ momenta are not a good quantum number for the interacting particles during the collision. 3. A significant time after the collision, particles are measured in momentum eigenstates. Momentum of each particle has stabilized and is again a good quantum number a long time after the collision. Theorem: A necessary and sufficient condition for the q_j to be good is that O commutes with the Hamiltonian H. A proof of this theorem is given below. Note that the theorem holds even if the spectrum is continuous; the proof is slightly more difficult (but no more illuminating) in that case.


Proof for Discrete Spectrum

We will work in the Heisenberg picture. Let O be a hermitian operator and let \psi_j be a complete, orthonormal basis of eigenstates with eigenvalues q_j. The unitary time-translation operator is U(t) = e^ and O(t) = U(t)^\dagger O U(t) so that \dot(t) = i/\hbar ,O(t)/math>. The main observation behind the proof is \fracq_j(t) = (i /\hbar )\langle \psi_j , ,O(t), \psi_j \rangle If O commutes with the hamiltonian then the right side vanishes and we get that the q_j are good quantum numbers. However, if we assume that the q_j are good quantum numbers then the left hand side vanishes for all j; since the \psi_j are complete, this implies that ,O= 0, which establishes the equivalence.


Ehrenfest Theorem and Good Quantum Numbers

The
Ehrenfest Theorem The Ehrenfest theorem, named after Paul Ehrenfest, an Austrian theoretical physicist at Leiden University, relates the time derivative of the expectation values of the position and momentum operators ''x'' and ''p'' to the expectation value of th ...
gives the rate of change of the
expectation value In probability theory, the expected value (also called expectation, expectancy, mathematical expectation, mean, average, or first moment) is a generalization of the weighted average. Informally, the expected value is the arithmetic mean of a l ...
of operators. It reads as follows: :\frac\langle A(t)\rangle = \left\langle\frac\right\rangle + \frac\langle (t),Hrangle Commonly occurring operators don't depend explicitly on time. If such operators commute with the
Hamiltonian Hamiltonian may refer to: * Hamiltonian mechanics, a function that represents the total energy of a system * Hamiltonian (quantum mechanics), an operator corresponding to the total energy of that system ** Dyall Hamiltonian, a modified Hamiltonian ...
, then their expectation value remains constant with time. Now, if the system is in one of the common
eigenstates In quantum physics, a quantum state is a mathematical entity that provides a probability distribution for the outcomes of each possible measurement on a system. Knowledge of the quantum state together with the rules for the system's evolution i ...
of the operatorA (and H too), then the system remains in this eigenstate as time progresses. Any measurement of the quantity A 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, and will be elaborated in more detail below.


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
commute Commute, commutation or commutative may refer to: * Commuting, the process of travelling between a place of residence and a place of work Mathematics * Commutative property, a property of a mathematical operation whose result is insensitive to th ...
s with the
Hamiltonian Hamiltonian may refer to: * Hamiltonian mechanics, a function that represents the total energy of a system * Hamiltonian (quantum mechanics), an operator corresponding to the total energy of that system ** Dyall Hamiltonian, a modified Hamiltonian ...
H. This implies that we can have common eigenstates of A and H. Assume that our system is in one of these common eigenstates. If we measure of A, it will definitely yield an eigenvalue of A (the good quantum number). Also, it is a well-known result that an eigenstate of the Hamiltonian is a
stationary state A stationary state is a quantum state with all observables independent of time. It is an eigenvector of the energy operator (instead of a quantum superposition of different energies). It is also called energy eigenvector, energy eigenstate, ener ...
, which means that even if the system is left to evolve for some time before the measurement is made, it will still yield the same eigenvalue. Therefore, If our system is in a common eigenstate, its eigenvalues of A (good quantum numbers) won't change with time. Conclusion: If ,H0 and the system is in a common eigenstate of A and H, the eigenvalues of A (good quantum numbers) don'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, ,H0. But now the system is not in any of the common eigenstates of H and A. So the system must be in some linear combination of the basis formed by the 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, they are bound to yield the same result. In this case, a (weaker) statement of conservation holds: Using the
Ehrenfest Theorem The Ehrenfest theorem, named after Paul Ehrenfest, an Austrian theoretical physicist at Leiden University, relates the time derivative of the expectation values of the position and momentum operators ''x'' and ''p'' to the expectation value of th ...
, ,H0 doesn't explicitly depend on time: \frac\langle A\rangle = \left\langle\frac\right\rangle + \frac\langle ,Hrangle = 0
This says that the
expectation value In probability theory, the expected value (also called expectation, expectancy, mathematical expectation, mean, average, or first moment) is a generalization of the weighted average. Informally, the expected value is the arithmetic mean of a l ...
of A remains constant in time. When the measurement is made on identical systems again and again, it will generally yield different values, but the expectation value remains constant. This is a weaker conservation condition than the case when our system was a common eigenstate of A and H: The eigenvalues of A are not ensured to remain constant, only its expectation value. Conclusion: If ,H0, A doesn't explicitly depend on time and the system isn't in a common eigenstate of A and H, the expectation value of A is conserved, but the conservation of the eigenvalues of A is not ensured.


Analogy with Classical Mechanics

In
classical mechanics Classical mechanics is a physical theory describing the motion of macroscopic objects, from projectiles to parts of machinery, and astronomical objects, such as spacecraft, planets, stars, and galaxies. For objects governed by classi ...
, the total
time derivative A time derivative is a derivative of a function with respect to time, usually interpreted as the rate of change of the value of the function. The variable denoting time is usually written as t. Notation A variety of notations are used to denote th ...
of a physical quantity A is given as: : \frac = \frac + \ where the curly braces refer to Poisson bracket of A and H. This bears a striking resemblance to the
Ehrenfest Theorem The Ehrenfest theorem, named after Paul Ehrenfest, an Austrian theoretical physicist at Leiden University, relates the time derivative of the expectation values of the position and momentum operators ''x'' and ''p'' to the expectation value of th ...
. It implies that a physical quantity A is conserved if its Poisson Bracket with the
Hamiltonian Hamiltonian may refer to: * Hamiltonian mechanics, a function that represents the total energy of a system * Hamiltonian (quantum mechanics), an operator corresponding to the total energy of that system ** Dyall Hamiltonian, a modified Hamiltonian ...
vanishes and the quantity does not depend on time explicitly. This condition in
classical mechanics Classical mechanics is a physical theory describing the motion of macroscopic objects, from projectiles to parts of machinery, and astronomical objects, such as spacecraft, planets, stars, and galaxies. For objects governed by classi ...
is analogous to the condition in
quantum mechanics Quantum mechanics is a fundamental theory in physics that provides a description of the physical properties of nature at the scale of atoms and subatomic particles. It is the foundation of all quantum physics including quantum chemistr ...
for the conservation of an
observable In physics, an observable is a physical quantity that can be measured. Examples include position and momentum. In systems governed by classical mechanics, it is a real-valued "function" on the set of all possible system states. In quantum phy ...
(as implied by
Ehrenfest Theorem The Ehrenfest theorem, named after Paul Ehrenfest, an Austrian theoretical physicist at Leiden University, relates the time derivative of the expectation values of the position and momentum operators ''x'' and ''p'' to the expectation value of th ...
: 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 In quantum physics, a quantum state is a mathematical entity that provides a probability distribution for the outcomes of each possible measurement on a system. Knowledge of the quantum state together with the rules for the system's evolution i ...
of the
Hamiltonian Hamiltonian may refer to: * Hamiltonian mechanics, a function that represents the total energy of a system * Hamiltonian (quantum mechanics), an operator corresponding to the total energy of that system ** Dyall Hamiltonian, a modified Hamiltonian ...
. They are also called
stationary state A stationary state is a quantum state with all observables independent of time. It is an eigenvector of the energy operator (instead of a quantum superposition of different energies). It is also called energy eigenvector, energy eigenstate, ener ...
s. 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: :\hat H , \Psi\rangle=E_ , \Psi\rangle, where *, \Psi\rangle is a
quantum state In quantum physics, a quantum state is a mathematical entity that provides a probability distribution for the outcomes of each possible measurement on a system. Knowledge of the quantum state together with the rules for the system's evolution i ...
, which is a stationary state; *\hat H is the
Hamiltonian operator Hamiltonian may refer to: * Hamiltonian mechanics, a function that represents the total energy of a system * Hamiltonian (quantum mechanics), an operator corresponding to the total energy of that system ** Dyall Hamiltonian, a modified Hamiltonia ...
; *E_ is the
energy eigenvalue A stationary state is a quantum state with all observables independent of time. It is an eigenvector of the energy operator (instead of a quantum superposition of different energies). It is also called energy eigenvector, energy eigenstate, ener ...
of the state , \Psi\rangle. The evolution of the state ket is governed by the
Schrödinger Equation The Schrödinger equation is a linear partial differential equation that governs the wave function of a quantum-mechanical system. It is a key result in quantum mechanics, and its discovery was a significant landmark in the development of th ...
: :i\hbar\frac , \Psi\rangle = E_, \Psi\rangle It gives the time evolution of the state of the system as: :, \Psi(t)\rangle = e^, \Psi(0)\rangle


Examples


The hydrogen atom

In non-relativistic treatment, l and s are good quantum numbers but in relativistic quantum mechanics they are no longer good quantum numbers as L and S do not commute with H (in Dirac theory). J=L+S is a good quantum number in relativistic quantum mechanics as J commutes with H.


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 Hamiltonian may refer to: * Hamiltonian mechanics, a function that represents the total energy of a system * Hamiltonian (quantum mechanics), an operator corresponding to the total energy of that system ** Dyall Hamiltonian, a modified Hamiltonian ...
are the orbital angular momentum, spin angular momentum, the sum of the spin angular momentum and orbital angular momentum, and the z components of the above angular momenta. Thus, the good quantum numbers in this case, (which are the eigenvalues of these observables) are l, j, m_\text , m_s, m_j. We have omitted s, 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 l, j, m_\text , m_s, m_j 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 : 1. A certain number of good quantum numbers can be used to specify uniquely a certain
quantum state In quantum physics, a quantum state is a mathematical entity that provides a probability distribution for the outcomes of each possible measurement on a system. Knowledge of the quantum state together with the rules for the system's evolution i ...
only when the
observable In physics, an observable is a physical quantity that can be measured. Examples include position and momentum. In systems governed by classical mechanics, it is a real-valued "function" on the set of all possible system states. In quantum phy ...
s corresponding to the good quantum numbers form a CSCO. 2. If the
observable In physics, an observable is a physical quantity that can be measured. Examples include position and momentum. In systems governed by classical mechanics, it is a real-valued "function" on the set of all possible system states. In quantum phy ...
s 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
observable In physics, an observable is a physical quantity that can be measured. Examples include position and momentum. In systems governed by classical mechanics, it is a real-valued "function" on the set of all possible system states. In quantum phy ...
s 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 hydrogen atom, the L^2, J^2 , L_z , J_z don't form a commuting set. But n, l, m_\text, m_s are the quantum numbers of a CSCO. So, are in this case, they form a set of good quantum numbers. Similarly, n, l, j, m_\text too form a set of good quantum numbers.


The hydrogen atom: spin-orbit interaction included

If the spin orbit interaction is taken into account, we have to add an extra term in
Hamiltonian Hamiltonian may refer to: * Hamiltonian mechanics, a function that represents the total energy of a system * Hamiltonian (quantum mechanics), an operator corresponding to the total energy of that system ** Dyall Hamiltonian, a modified Hamiltonian ...
which represents the
magnetic dipole In electromagnetism, a magnetic dipole is the limit of either a closed loop of electric current or a pair of poles as the size of the source is reduced to zero while keeping the magnetic moment constant. It is a magnetic analogue of the electric ...
interaction energy. :\Delta H_\text =-\boldsymbol\cdot\boldsymbol. Now, the new Hamiltonian with this new \Delta H_\text term doesn't
commute Commute, commutation or commutative may refer to: * Commuting, the process of travelling between a place of residence and a place of work Mathematics * Commutative property, a property of a mathematical operation whose result is insensitive to th ...
with \boldsymbol and \boldsymbol; but it does commute with L2, S2 and \boldsymbol , which is the
total angular momentum In quantum mechanics, the total angular momentum quantum number parametrises the total angular momentum of a given particle, by combining its orbital angular momentum and its intrinsic angular momentum (i.e., its spin). If s is the particle's sp ...
. In other words, m_\text, m_s are no longer good quantum numbers, but l, s, j , m_\text are. And since, good quantum numbers are used to label the
eigenstates In quantum physics, a quantum state is a mathematical entity that provides a probability distribution for the outcomes of each possible measurement on a system. Knowledge of the quantum state together with the rules for the system's evolution i ...
, the relevant formulae of interest are expressed in terms of them. For example, the spin-orbit interaction energy is given by :\Delta H_\text= (j(j+1) - l(l+1) -s(s+1)) where :\beta = \beta (n,l) = Z^4g_\text\mu_\text^2 As we can see, the above expressions contain the good quantum numbers, namely l,s, j


See also

*
Complete set of commuting observables In quantum mechanics, a complete set of commuting observables (CSCO) is a set of commuting operators whose common eigenvectors can be used as a basis to express any quantum state. In the case of operators with discrete spectra, a CSCO is a set of c ...
*
Hamiltonian (quantum mechanics) Hamiltonian may refer to: * Hamiltonian mechanics, a function that represents the total energy of a system * Hamiltonian (quantum mechanics), an operator corresponding to the total energy of that system ** Dyall Hamiltonian, a modified Hamiltoni ...
*
Stationary state A stationary state is a quantum state with all observables independent of time. It is an eigenvector of the energy operator (instead of a quantum superposition of different energies). It is also called energy eigenvector, energy eigenstate, ener ...
* Constant of motion * Quantum number *
Measurement in quantum mechanics In quantum physics, a measurement is the testing or manipulation of a physical system to yield a numerical result. The predictions that quantum physics makes are in general probabilistic. The mathematical tools for making predictions about what m ...
*
Ehrenfest theorem The Ehrenfest theorem, named after Paul Ehrenfest, an Austrian theoretical physicist at Leiden University, relates the time derivative of the expectation values of the position and momentum operators ''x'' and ''p'' to the expectation value of th ...
* Operator (physics)


References

{{Reflist Quantum mechanics