Most recent edit on 2008-08-13 03:39:14 by CharlesFrancis
Additions:
← Observable Quantities ↑ →
Deletions:
← Observable Quantities →
Edited on 2008-08-12 07:29:08 by CharlesFrancis
No differences.
Edited on 2008-08-12 02:44:41 by CharlesFrancis
Additions:
To say that an observable operator has a well defined value in a given state, a measurement should necessarily yield that value as the expectation of the operator. This is weaker than the projection postulate, which requires an eigenstate (in which the value is trivially given by the expectation).
Deletions:
To say that an observable operator has a well defined value in a given state, a measurement should necessarily yield that value as the expectation of the operator.
This is weaker than the projection postulate, which requires an eigenstate (in which the value is trivially given by the expectation).
Edited on 2008-08-12 02:35:55 by CharlesFrancis
Additions:
Since only a general principle has been used that it is possible to measure position, it is necessary to discuss other measurable quantities, or observables». I will be assume that all observables are a product of physical laws arising from interactions between particles. To describe a physically meaningful quantity, it will be required that an operator can be constructed from the operators describing the physical interactions of matter. A full analysis of a given measurement would require that the measurement apparatus as well as the system being measured be treated as an interacting multiparticle system. This must wait until a complete model of interactions between particles has been developed. Here general considerations are discussed, on the assumption that interactions will be described by linear operators and that a measurement is a physical process which can, at least in principle, be described as a combination of interaction operators.
The projection postulate was due originally to Von Neumann, for observables with discrete values. It was extended by Lüders to continuous observables. It is too restrictive to describe all numerical quantities used in the classical description of nature, and will be relaxed after a discussion of expectations.
Deletions:
Since only a general principle has been used that it is possible to measure position, it is necessary to discuss other measurable quantities, or observables». I will be assume that all observables are a product of physical laws arising from interactions between particles. A full analysis of a given measurement would require that the measurement apparatus as well as the system being measured be treated as an interacting multiparticle system. This must wait until a complete model of interactions between particles has been developed. Here general considerations are discussed, on the assumption that interactions will be described by linear operators and that a measurement is a physical process which can, at least in principle, be described as a combination of interaction operators.
The projection postulate was due originally to Von Neumann, for observables with discrete values and extended by Lüders to continuous observables. It is too restrictive to describe all numerical quantities used in the description of nature, and will be relaxed after a discussion of expectations.
Postulate: Classical quantities are given by the expectation of corresponding Hermitian operators (irrespective of whether the state is an eigenstate).
For physically meaningful quantities it will also be required that a Hermitian operator can be constructed from the operators describing the physical interactions of matter.
Edited on 2008-08-12 02:05:15 by CharlesFrancis
Additions:
The projection postulate was due originally to Von Neumann, for observables with discrete values and extended by Lüders to continuous observables. It is too restrictive to describe all numerical quantities used in the description of nature, and will be relaxed after a discussion of expectations.
Deletions:
The projection postulate was due originally to Von Neumann, for observables with discrete values and extended by Lüders to continuous observables. It is too restrictive to describe all Myrddin, Son of Morfryn quantities of nature, and will be relaxed after a discussion of expectations.
, but it is not necessary to have an eigenstate of an observable operator for the classical observable to exist with a definite value.
Edited on 2008-08-12 02:02:02 by CharlesFrancis
Additions:
This definition ties down the relative magnitudes of the complex numbers, a and b, in rule II, and makes precise the weighting in logical OR.
The projection postulate was due originally to Von Neumann, for observables with discrete values and extended by Lüders to continuous observables. It is too restrictive to describe all Myrddin, Son of Morfryn quantities of nature, and will be relaxed after a discussion of expectations.
If a configuration of matter corresponds to an eigenstate» of an observable operator then the value of that observable exists independently of observation and is given by the corresponding eigenvalue». For physically meaningful observable quantities it will also be required that the Hermitian operator can be constructed, in some state, from the operators describing the physical interactions of matter. In a measurement of an observable, K, obeying the projection postulate, the probability that operators describing the interactions combine to 
is
More generally, we have that if the state at time t is known to be 
, then the probability that a measurement at time t will give the result 
is given by
as the truth value of the proposition 
.
, but it is not necessary to have an eigenstate of an observable operator for the classical observable to exist with a definite value.
Classical Quantities
The projection postulate is required if the results of measurement are to be used to name states in Hilbert space, but classical quantities can also be defined from Hermitian operators when this is not the case. For an observable obeying the projection postulate, the corresponding measurement necessarily creates an eigenstate. More generally, a real value valued quantity is defined by the expectation of any hermitian operator
To say that an observable operator has a well defined value in a given state, a measurement should necessarily yield that value as the expectation of the operator.
Postulate: Classical quantities are given by the expectation of corresponding Hermitian operators (irrespective of whether the state is an eigenstate).
This is weaker than the projection postulate, which requires an eigenstate (in which the value is trivially given by the expectation).
For physically meaningful quantities it will also be required that a Hermitian operator can be constructed from the operators describing the physical interactions of matter.
Time-Energy Uncertainty
If a configuration of matter corresponds to an eigenstate» of an observable operator then the value of that observable exists independently of observation and is given by the corresponding eigenvalue», but it is not necessary to have an eigenstate of an observable operator for the classical observable to exist with a definite value.
For physically meaningful observable quantities it will also be required that the Hermitian operator can be constructed, in some state, from the operators describing the physical interactions of matter.
In a measurement of an observable, K, obeying the projection postulate, the probability that operators describing the interactions combine to Ki is
More generally, we have that if the state at time t is known to be , then the probability that a measurement at time t will give the result 
is given by
as the truth value of the proposition .
Time
Edited on 2008-08-09 02:56:31 by CharlesFrancis
Additions:
Postulate: The classical value of an observable quantity is given by the expectation of the corresponding Hermitian operator (irrespective of whether the state is an eigenstate).
For physically meaningful observable quantities it will also be required that the Hermitian operator can be constructed, in some state, from the operators describing the physical interactions of matter.
Deletions:
The classical value of an observable quantity is given by the expectation of the corresponding Hermitian operator (irrespective of whether the state is an eigenstate).
Edited on 2008-08-09 02:51:07 by CharlesFrancis
Additions:
The classical value of an observable quantity is given by the expectation of the corresponding Hermitian operator (irrespective of whether the state is an eigenstate).
Deletions:
The classical value of an observable quantity is given by the expectation of the corresponding observable operator (irrespective of whether the state is an eigenstate).
Edited on 2008-08-09 02:48:46 by CharlesFrancis
Additions:
The projection postulate was due originally to Von Neumann, for observables with discrete values and extended by Lüders to continuous observables. It is too restrictive to describe all the measureable quantities of nature, and will be relaxed after a discussion of expectations.
If a configuration of matter corresponds to an eigenstate» of an observable operator then the value of that observable exists independently of observation and is given by the corresponding eigenvalue», but it is not necessary to have an eigenstate of an observable operator for the classical observable to exist with a definite value.
The classical value of an observable quantity is given by the expectation of the corresponding observable operator (irrespective of whether the state is an eigenstate).
In a measurement of an observable, K, obeying the projection postulate, the probability that operators describing the interactions combine to Ki is
Deletions:
If a configuration of matter corresponds to an eigenstate» of an observable operator then the value of that observable exists independently of observation and is given by the corresponding eigenvalue».
In a measurement of K, the probability that operators describing the interactions combine to Ki is
Edited on 2008-08-06 06:03:47 by CharlesFrancis
Additions:
Time
Hilbert space has been set up to describe measurement at particular time. So time is a parameter, not an operator. The time-energy uncertainty relation. In the uncertainty relation for time-energy,
t does not refer to the time parameter. An intuitive interpretation is that a state which exists for only a short time, Δt, cannot have a definite energy, since to have a definite energy the frequency of the state needs to be defined over many cycles, the reciprocal of the required accuracy.
For example, in spectroscopy, excited states have a finite lifetime. By the time-energy uncertainty principle, they do not have a definite energy, and each time they decay the energy they release is slightly different. The average energy of the outgoing photon has a peak at the theoretical energy of the state, but the distribution has a finite width called the natural linewidth. Fast-decaying states have a broad linewidth, while slow decaying states have a narrow linewidth. Similarly, in particle physics, the faster a particle decays, the less certain is its mass.
In 1936, Dirac gave a precise definition and derivation of the time-energy uncertainty relation in a relativistic quantum theory of "events", in which particles follow trajectories parametrized independently by a different proper time. There is uncertainty in the time at which any event, or interaction for the particle, takes place. This “many-times” formulation inspired Shin-Ichiro Tomonaga»'s development of quantum electrodynamics.
Deletions:
The uncertainty relation for time-energy,
is more subtle. Hilbert space has been set up to describe measurement at particular time. So time is a parameter, not an operator. The time-energy uncertainty relation can be interpreted in a number of ways, and is not discussed on this website.
Edited on 2008-08-06 04:14:54 by CharlesFrancis
Additions:
Classical probability theory describes situations in which every parameter exists, but some are not known. Probabilistic results come from different values taken by unknown parameters. We have a similar situation here, but now the unknowns are not describable as parameters. We assume no relationships between particles bar those generated by physical interaction. An experiment is described as a large configuration of particles incorporating the measuring apparatus as well as the process being measured. The configuration has been partially determined by setting up the experimental apparatus, reducing the possibilities to those with definite outcomes to the measurement, but the fine details of the configuration of matter is unknown. It is impossible, even in principle, to determine every detail of the configuration since the determination of each detail requires measurement, which in turn requires a larger apparatus containing new unknowns in the configuration of particles. Thus, there is always a lack of determination of initial conditions leading to randomness in the outcome, whether or not there is a fundamental indeterminism in Nature.
In typical treatments of quantum mechanics based on canonical quantisation», 3-vector numerical quantities, x and p, representing position and momentum are simply replaced with operators x and p, and the canonical commutation relation, for a, b = 1, 2, 3,
Theorem: The position and momentum observables, x and p, obey the commutation relation,
This result, which is generally called the Heisenberg uncertainty principle, was first proved by Kennard, in 1927.
The uncertainty relation for time-energy,
is more subtle. Hilbert space has been set up to describe measurement at particular time. So time is a parameter, not an operator. The
time-energy uncertainty relation can be interpreted in a number of ways, and is not discussed on this website.
Deletions:
Classical probability theory describes situations in which every parameter exists, but some are not known. Probabilistic results come from different values taken by unknown parameters. We have a similar situation here, but now the unknowns are not describable as parameters. We assume no relationships between particles bar those generated by physical interaction. An experiment is described as a large configuration of particles incorporating the measuring apparatus as well as the process being measured. The configuration has been partially determined by setting up the experimental apparatus, reducing the possibilities to those with definite outcomes to the measurement, but the fine details of the configuration of matter is unknown. It is impossible, even in principle, to determine every detail of the configuration since the determination of each detail requires measurement, which in turn requires a larger apparatus containing new unknowns in the configuration of particles. Thus, there is always a lack of determination of initial conditions leading to randomness in the outcome, whether or not there is a fundamental indeterminism in nature.
In typical treatments of quantum mechanics based on canonical quantisation», numerical quantities, x and p, representing position and momentum are simply replaced with operators x and p, and the canonical commutation relation,
Theorem: The position and momentum observables, x and p, obey the commutation relation
This result, which is generally called the Heisenberg uncertainty principle, was first proved by Kennard, in 1927. We read the uncertainty relation for time-energy from the time components, E = p0 and t = x0,
Edited on 2008-05-02 04:56:38 by CharlesFrancis
Additions:
Definition: An effectively measured state is one which a definite measurement result can, in principle, be calculated from other measurements by means of known physical law.
Edited on 2008-04-26 01:25:30 by CharlesFrancis
Additions:
Deletions:
Edited on 2008-04-26 01:23:16 by CharlesFrancis
Additions:
is imposed. This procedure is usually justified only because it leads to an empirically accurate theory. In the present treatment, quantum theory has been set up from fundamental principles. It is necessary to define the operators x and p and to show that they obey the canonical commutation relation. The definitions follow the general from for observable operators.
Proof: For any ket 
,
Hence,
So,
holds for all , as required.
Deletions:
is imposed. This procedure is usually justified only because it leads to an empirically accurate theory. In the present treatment, quantum theory has been set up from fundamental principles. It is necessary to define the operators x and p and to show that they obey the canonical commutation relation.
Proof: For any ket
Hence
This is true for all . The commutation relation follows immediately.
Edited on 2008-04-25 07:15:05 by CharlesFrancis
Additions:
In typical treatments of quantum mechanics based on canonical quantisation», numerical quantities, x and p, representing position and momentum are simply replaced with operators x and p, and the canonical commutation relation,
where the commutator between operators A and B is [A, B] = AB − BA.
Deletions:
In typical treatments of quantum mechanics based on canonical quantisation», numerical quantities, x and p, representing position and momentum are simply replaced with operators x and p, and the canonical commutation relation,
where the commutator between operators A and B is [A, B] = AB − BA.
Edited on 2008-04-25 06:47:32 by CharlesFrancis
Additions:
Theorem: The momentum observable, p, is given by
Deletions:
""<span class="math"><b>Theorem:</b> ; The momentum observable, <i>p</i>, is given by
Edited on 2008-04-25 06:31:12 by CharlesFrancis
Additions:
This is true for all . The commutation relation follows immediately.
Proof: Let 
and 
. Then, since A and B are Hermitian,
The general uncertainty relation follows by applying this result to 
and 
, observing that
Deletions:
This is true for all . The commutation relation follows immediately.
and . Then, since A and B are Hermitian,
The general uncertainty relation follows by applying this result to 
and 
, observing that
In typical treatments of quantum mechanics based on canonical quantisation», numerical quantities, x and p, representing position and momentum are simply replaced with operators x and p, and the canonical commutation relation,
is imposed. This procedure is usually justified only because it leads to an empirically accurate theory. In the present treatment, quantum theory has been set up from fundamental principles. It is necessary to define the operators x and p and to show that they obey the canonical commutation relation.
Definition: The position observable is
Definition: The momentum observable is
The notation overloads the symbols
x and
p, in accordance with standard practice. Care should be taken over whether
x and
p are operators (as on the LHS) or numerical quantities (as on the RHS).
As an operator, the position observable is a map
So
x maps the state with wave function
to the state with wave function
. In this treatment, wave functions are regarded as the coefficients of kets in a basis of position states. In some treatments wave functions are treated as a representation of Hilbert space. In this case states are identified with wave functions and the position observable is given by,
Theorem: The momentum observable, p, is given by
<img title="Differential form of the momentum observable" alt="Observables-53" src="images/observables/Observables-53g.gif" align="texttop" vspace="2">
**Proof:** For any <a href=http://www.teleconnection.info/rqg/FoundationsOfQuantumTheory#PlaneWaveStates>plane» wave state</a>, <img title="Momentum ket" alt="Observables-54" src="images/observables/Observables-54.gif" align="texttop" vspace="0">
<img title="Differentiate the wave function directly" alt="Observables-55" src="images/observables/Observables-55.gif" align="texttop" vspace="3">
<img title="Use the resolution of unity" alt="Observables-56" src="images/observables/Observables-56.gif" align="texttop" vspace="1">
<img title="Use the resolution of unity in momentum space" alt="Observables-57" src="images/observables/Observables-57.gif" align="texttop" vspace="1">
<img title="Use the inner product of plane wave states" alt="Observables-58" src="images/observables/Observables-58.gif" align="texttop" vspace="3">
This is true for all states, <img title="Momentum ket" alt="Observables-59" src="images/observables/Observables-59.gif" align="texttop" vspace="0">, in a basis, so the theorem holds for all states by linearity.
As a mapping on the representation of wavefunctions,
<img title="Momentum operator as a map on wave functions" alt="Observables-60" src="images/observables/Observables-60.gif" align="texttop" vspace="3">
and we write
<img title="Momentum operator as a map on wave functions" alt="Observables-61" src="images/observables/Observables-61.gif" align="texttop" vspace="3">
<<span class="math"><b>Theorem:</b> ; The position and momentum observables, <i>x</i> and <i>p</i>, obey the commutation relation
<img title="Canonical commutation relation" alt="Observables-62" src="images/observables/Observables-62g.gif" align="texttop" vspace="3">
<span class="math">or, in conventional units,
<img title="Canonical commutation relation in conventional units" alt="Observables-63" src="images/observables/Observables-63g.gif" align="texttop" vspace="3">
**Proof:** For any ket <img title="any ket" alt="Observables-64" src="images/observables/Observables-64.gif" align="texttop" vspace="0">
<img title="evaluate the product" alt="Observables-65" src="images/observables/Observables-65.gif" align="texttop" vspace="3">
Hence
<img title="Evaluate" alt="Observables-66" src="images/observables/Observables-66.gif" align="texttop" vspace="3">
<img title="the inner product is a Kronecker delta" alt="Observables-67" src="images/observables/Observables-67.gif" align="texttop" vspace="3">
<img title="differentiate using the product rule" alt="Observables-68" src="images/observables/Observables-68.gif" align="texttop" vspace="3">
<img title="use the resolution of unity" alt="Observables-68" src="images/observables/Observables-68.gif" align="texttop" vspace="3">""
This is true for all . The commutation relation follows immediately.
The general uncertainty relation can be applied to any two observables. For example, the uncertainty in position-momentum is
In conventional units
Deletions:
The general uncertainty relation can be applied to any two observables. For example, the position observable is
and the momentum observable is
as may be seen by differentiating the coefficients for plane wave states. Then we find, by applying the product rule for differentiation, for uncertainty in position-momentum,
This gives the uncertainty relation in natural units. In conventional units we have
Edited on 2008-03-01 01:02:00 by CharlesFrancis
Additions:
It has been said that measurement of time and position sufficient for the study of all physical quantities. For example, a classical measurement of velocity may be reduced to a time trial over a measured distance, and a typical measurement of momentum of a particle involves plotting its path in a bubble chamber, being a set of positions over a time interval. In the most general case, measurement of position might refer to the position of particles other than the one under study, such as the position of a pointer. The language of quantum logic is here found sufficient to discuss probabilities for the results of any measurement where this is so.
← Observable Quantities →
It has been said that measurement of time and position sufficient for the study of all physical quantities. For example, a classical measurement of velocity may be reduced to a time trial over a measured distance, and a typical measurement of momentum of a particle involves plotting its path in a bubble chamber, being a set of positions over a time interval. In the most general case, measurement of position might refer to the position of particles other than the one under study, such as the position of a pointer. The language of Quantum logic is here found sufficient to discuss probabilities for the results of any measurement where this is so.
Probability Interpretation
To make the language precise, we need to attach numerical values to the complex numbers introduced in rule II. To determine a complex number we must determine its magnitude and phase. Phase contains information on the evolution of kets, and is found from relativistic considerations. Magnitude will be determined from probability. It only makes sense to talk about probability when we are actually going to do a measurement. When we are actually going to do the measurement, a statement about hypothetical measurement, in the subjunctive mood, automatically becomes a statement about real measurement, in the future tense. This being the case, truth values for hypothetical results must become truth values for future events, i.e. probabilities. We require that the language generates the probability, 
, of getting the result, x, when a measurement of position is actually going to be performed on the state 
.
Two conditions are required of a probability density function». It must be greater than or equal to zero for all values of its argument, and it must have an area under its graph equal to 1 — or, in 3 dimensions, a hyper-volume under its graph of 1. Using the integral representation of the resolution of unity,
So, for a positive definite inner product, the function,
has the property of a probability density.
Definition: The magnitudes of the coefficients, 
, are such that, for all x,
is the conditional probability that a measurement of position will return the result x, given the information from a prior measurement contained in the ket 
.
This definition ties down the relative magnitudes of the complex numbers, a and b, in rule II, and makes precise the weighting in logical OR.
Measurement
Since only a general principle has been used that it is possible to measure position, it is necessary to discuss other measurable quantities, or observables». I will be assume that all observables are a product of physical laws arising from interactions between particles. A full analysis of a given measurement would require that the measurement apparatus as well as the system being measured be treated as an interacting multiparticle system. This must wait until a complete model of interactions between particles has been developed. Here general considerations are discussed, on the assumption that interactions will be described by linear operators and that a measurement is a physical process which can, at least in principle, be described as a combination of interaction operators.
Definition: A measurement of a physical quantity is any physical process such that a determination of the quantity is possible in principle.
This is a very broad definition of measurement, and does not require the intervention of a human observer. In classical physics, complete knowledge of the situation at a particular time contains sufficient information to determine the situation at any other time. When classical laws apply, and the state at some time is known from measurement, the state at all times can be calculated, at least in principle. For example, we can reasonably assume that Venus continues on its orbit while it is on the far side of the Sun and cannot be observed. Experience tells us that when Venus reappears it will be observed in precisely the position found by calculation from prior measurements. It is unreasonable to think that the same calculations do not hold while Venus is not observed. When classical laws apply, and the state has been measured at some time, then the state can be said to be effectively measured at any time, whether or not it is actually observed, and whether or not a calculation is actually performed.
Quantum Measurement
Measurement has two effects on the state of a particle, altering it due to the interaction of the apparatus with the particle, and also changing the information we have about the state. New information causes a change of state even in the absence of physical change because the state, or ket, is just a label for available information. The ket changes in part due to the effect of the apparatus on the particle, and in part due to the change in a conditional probability when the condition becomes known. This inverts the measurement problem»; collapse represents a change in information due to a new measurement but Schrödinger's equation» requires explanation — interference patterns are real. The requirement for a wave equation will be found in the next section, Time Evolution.
Classical probability theory describes situations in which every parameter exists, but some are not known. Probabilistic results come from different values taken by unknown parameters. We have a similar situation here, but now the unknowns are not describable as parameters. We assume no relationships between particles bar those generated by physical interaction. An experiment is described as a large configuration of particles incorporating the measuring apparatus as well as the process being measured. The configuration has been partially determined by setting up the experimental apparatus, reducing the possibilities to those with definite outcomes to the measurement, but the fine details of the configuration of matter is unknown. It is impossible, even in principle, to determine every detail of the configuration since the determination of each detail requires measurement, which in turn requires a larger apparatus containing new unknowns in the configuration of particles. Thus, there is always a lack of determination of initial conditions leading to randomness in the outcome, whether or not there is a fundamental indeterminism in nature.
When we do a measurement, K, we get a definite result, a terminating decimal or n-tuple of terminating decimals read off the measurement apparatus. Let the possible results be ki for i = 1, …, n . We assume that the dimension of H1 is greater than n; this must be so if all measurements are reducible to measurements of position, and can be ensured through an appropriate choice of lattice at finer than the resolution of measurement. Each physical state is associated with a ket, labelled by the measurement result, so that if the measured result is ki then the state is 
. The empirical determination of 
as a member of H1 requires that we draw from experimental data the value of the inner product 
for an arbitrary state, 
. Without loss of generality 
and 
are normalised. By assumption, measurement of K is reducible to a set of measurements of position, so that each ki is in one to one correspondence with the positions yi of one or more particles used for the measurement (e.g. yi may be the positions of one or more pointers). Then,
is the probability that a measurement of K has result ki, given the initial state 
in H1.
If the result is ki, it is definitely ki and cannot be given as kj with i ≠ j by another measurement of K at the same time. So,
Measurement with result, ki, implies a physical action on a system and is represented by the action of an operator, Ki, on Hilbert space. If a quantity is measurable we require that there is an element of physical reality associated with its measurement, meaning that, in measurement, the configuration of particles necessarily becomes such that the quantity has a well defined value. In practice, this means that, in the limit in which the time between two measurements goes to zero, a second measurement of the quantity necessarily gives the same result as the first. It follows that Ki is a projection operator», and has the form
.
Definition: A projection operator, K, is a linear operator such that K2 = K.
Definition: The projection postulate states that the effect of a measurement is equivalent to the action of a projection operator.
Observable Operators
The expectation» of the result from a measurement of K, given an initial normalised state, 
in H1, is
Definition: An observable» is a Hermitian operator with the form,
The effect of K on the state 
is to leave it unchanged.
ki is the value of K in the state 
.
If a configuration of matter corresponds to an eigenstate» of an observable operator then the value of that observable exists independently of observation and is given by the corresponding eigenvalue».
In a measurement of K, the probability that operators describing the interactions combine to Ki is
.
can be understood as a classical probability function, where the random variable runs over the set of projection operators, Ki, corresponding to the outcomes of the measurement. The physical interpretation is that each Ki represents a set of unknown configurations of particle interactions in measurement, namely that set of configurations leading to the result ki.
More generally, we have that if the state at time t is known to be , then the probability that a measurement at time t will give the result is given by
Definition: To complete the structure of a many valued logic, we regard
as the truth value of the proposition .
The Uncertainty Principle
The uncertainty principle» can be derived from the theory while the theory cannot be derived from it. Strictly, this makes it a theorem, not a principle. Uncertainty in an observable is defined in the same way as variance» in statistics. It is a prediction of the root mean squared difference between observed values and the expected value».
Definition: The uncertainty, ΔK in an observable operator, K, is
Theorem: Any two observables A and B obey the general uncertainty relation,
where the commutator between operators A and B is [A, B] = AB − BA.
Proof: Let and . Then, since A and B are Hermitian,
by the Cauchy-Schwarz inequality. Thus,
The general uncertainty relation follows by applying this result to and , observing that
.
The general uncertainty relation can be applied to any two observables. For example, the position observable is
and the momentum observable is
as may be seen by differentiating the coefficients for plane wave states. Then we find, by applying the product rule for differentiation, for uncertainty in position-momentum,
This gives the uncertainty relation in natural units. In conventional units we have
This result, which is generally called the Heisenberg uncertainty principle, was first proved by Kennard, in 1927. We read the uncertainty relation for time-energy from the time components, E = p0 and t = x0,
Observables ↑ Evolution of Quantum States →
, 
, then k is an eigenvalue and 
is an eigenstate of K.
