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 chemistry, q ...
, given a particular
Hamiltonian and an
operator
Operator may refer to:
Mathematics
* A symbol indicating a mathematical operation
* Logical operator or logical connective in mathematical logic
* Operator (mathematics), mapping that acts on elements of a space to produce elements of another ...
with corresponding
eigenvalues
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 b ...
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
, the
are said to be good quantum numbers if every eigenvector
remains an eigenvector of
''with the same eigenvalue'' as time evolves.
In other words, the eigenvalues
are good quantum numbers if the corresponding operator
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
to be good is that
commutes with the Hamiltonian
.
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
be a hermitian operator and let
be a complete, orthonormal basis of eigenstates with eigenvalues
. The unitary time-translation operator is
and
so that
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, 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 in t ...
of the operator
A (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
Operator may refer to:
Mathematics
* A symbol indicating a mathematical operation
* Logical operator or logical connective in mathematical logic
* Operator (mathematics), mapping that acts on elements of a space to produce elements of another ...
which
commutes with the
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 classical ...
, 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
In mathematics and classical mechanics, the Poisson bracket is an important binary operation in Hamiltonian mechanics, playing a central role in Hamilton's equations of motion, which govern the time evolution of a Hamiltonian dynamical system. T ...
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
In mathematics and classical mechanics, the Poisson bracket is an important binary operation in Hamiltonian mechanics, playing a central role in Hamilton's equations of motion, which govern the time evolution of a Hamiltonian dynamical system. T ...
with the
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 classical ...
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 chemistry, q ...
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 phys ...
(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
In mathematics and classical mechanics, the Poisson bracket is an important binary operation in Hamiltonian mechanics, playing a central role in Hamilton's equations of motion, which govern the time evolution of a Hamiltonian dynamical system. T ...
is replaced by
commutator
In mathematics, the commutator gives an indication of the extent to which a certain binary operation fails to be commutative. There are different definitions used in group theory and ring theory.
Group theory
The commutator of two elements, ...
)
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 in t ...
of the
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 in ...
, 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 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 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
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 b ...
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 in ...
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 phys ...
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 phys ...
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 phys ...
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 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 with
\boldsymbol and
\boldsymbol; but it does commute with L
2, S
2 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 s ...
. 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 in t ...
, 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 co ...
*
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 Hamilton ...
*
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 In mechanics, a constant of motion is a quantity that is conserved throughout the motion, imposing in effect a constraint on the motion. However, it is a ''mathematical'' constraint, the natural consequence of the equations of motion, rather than ...
*
Quantum number
In quantum physics and chemistry, quantum numbers describe values of conserved quantities in the dynamics of a quantum system. Quantum numbers correspond to eigenvalues of operators that commute with the Hamiltonian—quantities that can b ...
*
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)
In physics, an operator is a function over a space of physical states onto another space of physical states. The simplest example of the utility of operators is the study of symmetry (which makes the concept of a group useful in this context). B ...
References
{{Reflist
Quantum mechanics