Heisenberg's uncertainty relation

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Chart of the Heisenberg Indetermination Principle.

In quantum mechanics, the Heisenberg uncertainty relation or uncertainty principle establishes the impossibility of certain pairs of observable and complementary physical quantities being known with arbitrary precision. Succinctly, it affirms that it is not possible to determine, in terms of quantum physics, simultaneously and with arbitrary precision, certain pairs of physical variables, such as the position and the linear momentum (amount of movement) of a given object. In other words, the more certainty one seeks in determining the position of a particle, the less one knows its linear momentum and, therefore, its mass and velocity. This principle was stated by the German theoretical physicist Werner Heisenberg in 1927.

There is currently a slight error caused by classical thinking so deeply rooted in human reasoning, there is a tendency to believe that indeterminacy is due to experimental intervention when measuring a property. However, what the uncertainty principle suggests is that the properties of the particle are in a state of superposition and therefore have different values of position and linear momentum attributed to them at the same time. In the intervention, when measuring, we force one of the magnitudes to take a value, collapsing its wave function, and thus giving us a precise result for it, thus indeterminately increasing the indeterminacy in the other measurement.

The uncertainty principle has no classical analogue and defines one of the fundamental differences between classical physics and quantum physics. From a logical point of view it is a consequence of current axioms of quantum mechanics and therefore it strictly follows from them.

Qualitative explanation of the uncertainty principle

The “informative” explanation of the uncertainty principle states that dynamic variables such as position, angular momentum, linear momentum, etc. They are defined in an operational way, that is, in terms relative to the experimental procedure by means of which they are measured: the position will be defined with respect to a determined reference system, defining the measurement instrument used and the how such an instrument is used (for example, measuring with a ruler the distance from such a point to the references).

However, when one examines the experimental procedures by which such variables could be measured, it turns out that the measurement will always end up perturbed. Indeed, if, for example, we think about what would be the measurement of the position and speed of an electron, to carry out the measurement (in order to be able to "see" the electron in some way) it is necessary for a photon of light to collide with the electron, with which it is modifying its position and speed; that is, by the very act of making the measurement, the experimenter modifies the data in some way, introducing an error that is impossible to reduce to zero, no matter how perfect our instruments are.

Note that if the position is measured, determining the disturbance that the particle generates in the gravitational field that surrounds it, the error can be reduced to zero. Because every particle is affected in different measures by the fields generated by others.

This qualitative description of the principle, while not being totally incorrect, is misleading in that it omits the main aspect of the uncertainty principle: the uncertainty principle sets the limit of applicability of classical physics. Classical physics conceives physical systems described by means of perfectly defined variables in time (velocity, position,...) and that in principle can be known with the precision that is desired. Although in practice it would be impossible to determine the position of a particle with infinitesimal precision, classical physics conceives such precision as achievable: it is possible and perfectly conceivable to affirm that this or that particle, at the exact moment of time 2 s, was in the exact position 1.57 m. Instead, the uncertainty principle, by stating that there is a fundamental limit to the precision of the measurement, is actually indicating that if a real physical system is described in terms of classical physics, then an approximation is being made, and the uncertainty ratio tells us the quality of that approximation.

For cultural and educational reasons, people often face the principle of uncertainty for the first time being conditioned by the determinism of classical physics. In it, position $x$ of a particle can be defined as a continuous function in time, ${displaystyle x=x(t)}$ . If the mass of that particle is $m$ and moves at speeds sufficiently lower than that of light, then the linear moment of the particle is defined as mass by speed, being the speed the first derived in the time of position: ${displaystyle p=m{frac {dx}{dt}}}$ .

That said, in view of the usual explanation of the principle of uncertainty, it might be tempting to believe that the relationship of uncertainty simply establishes a limitation on our ability to measure that prevents us from knowing arbitrarily the initial position ${displaystyle x(0)}$ and the initial linear moment ${displaystyle p(0)}$ . It happens if we could meet ${displaystyle x(0)}$ and ${displaystyle p(0)}$ then classical physics would offer us the position and speed of the particle at any other moment; the general solution of the equations of movement will depend invariably on ${displaystyle x(0)}$ and ${displaystyle p(0)}$ . That is, solving the equations of the movement leads to a family or set of trajectories dependent on ${displaystyle x(0)}$ and ${displaystyle p(0)}$ according to what value they take ${displaystyle x(0)}$ and ${displaystyle p(0)}$ it will have a trajectory within that family or another, but the very resolution of the equations limits the number of trajectories to a particular set of them. As reasoned, according to the principle of uncertainty ${displaystyle x(0)}$ and ${displaystyle p(0)}$ You can't know exactly, so you can't get to know each other either. ${displaystyle x(t)}$ and ${displaystyle p(t)}$ at any other moment with arbitrary precision, and the trajectory that the particle will follow cannot be known in an absolutely accurate manner. This reasoning is, however, incorrect, because in it subdues the idea that, although ${displaystyle x(0)}$ and ${displaystyle p(0)}$ cannot be known exactly, it is possible to continue using the classic description under which a particle will follow a trajectory defined by the general solution of the equations of movement, introducing the added notion that the initial conditions ${displaystyle x(0)}$ and ${displaystyle p(0)}$ they can't be known in detail: that is, we can't know exactly what trajectory the particle is going to follow, but we will be accepting that, de facto, it will follow one.

This way of proceeding is, however, totally incorrect: the principle of uncertainty entails a complete deviation of classic conceptions, making the classical notion of trajectory must be discarded: ask which values are simultaneously ${displaystyle x(t)}$ and ${displaystyle p(t)}$ It's absurd. Thus, it could be paradoxical that a relationship of uncertainty should first be established in terms of position $x$ and linear moment $p$ to then claim that $x$ and $p$ which appear in such a relationship make no sense: if they make no sense, what sense can a relationship have to use them? It happens that, in quantum physics, it is possible to introduce a series of mathematical entities $x$ and $p$ that correspond in many respects with the classic position and moment. These entities are not, however, exactly the same as the classical position and moment: the principle of uncertainty simply indicates that if we interpret these entities as a linear position and moment - and therefore we interpret the movement in a classical form - then there is a fundamental limit in the accuracy with which these variables can be known; that is, if we try to introduce classic variables and try to interpret the movement in a classical way, the accuracy with which these limited variables can be specified is.

Consequences of the relationship of indeterminacy

This principle supposes a basic change in the nature of physics, since it goes from absolutely precise knowledge (in theory but not in practice), to knowledge based only on probabilities. Although due to the smallness of Planck's constant, in the macroscopic world quantum indeterminacy is almost always completely negligible, and the results of deterministic physical theories, such as the theory of relativity, remain valid in all practical cases of interest.

Particles, in quantum mechanics, do not follow defined trajectories. It is not possible to know exactly the value of all the physical magnitudes that describe the state of motion of the particle at any moment, but only a statistical distribution. Therefore it is not possible to assign a trajectory to a particle. Yes, it can be said that there is a certain probability that the particle is in a certain region of space at a certain moment.

The probabilistic nature of quantum mechanics is commonly considered to invalidate scientific determinism. However, there are several interpretations of quantum mechanics and not all reach this conclusion. As Stephen Hawking points out, quantum mechanics is itself deterministic, and it is possible that the apparent indeterminacy is due to the fact that there really are no positions and velocities of particles, only waves. Quantum physicists would then try to fit the waves to our preconceived ideas of positions and speeds. The inadequacy of these concepts would be the cause of the apparent unpredictability. Other deductible phenomena or connected with the Heisenberg uncertainty principle are:

Tunnel effect
Zero-point energy
Existence of virtual particles
Energy of vacuum and absence of absolute vacuum.
Hawking radiation and instability of black holes

Mathematical statement

If several identical copies of a system are prepared in a certain state, such as an atom, the measurements of the position and momentum will vary according to a certain probability distribution characteristic of the quantum state of the system. The measurements of the observable object will suffer standard deviation Δx of the position and momentum Δp. Then check the principle of indeterminacy that is expressed mathematically as:

${displaystyle Delta xcdot Delta pgeq {frac {hbar }{2}}}$

where the h is the constant of Planck (to simplify, ${displaystyle {frac {h}{2pi }}}$ usually writes as ${displaystyle hbar }$ )

The known value of Planck's constant is:

{displaystyle h=,,6,626 0693(11)times 10^{-34} {mbox{J}}cdot {mbox{s}},,=,,4,135 667 43(35)times 10^{-15} {mbox{eV}}cdot {mbox{s}}}

In classical systems physics this position-momentum indeterminacy does not manifest since it applies to quantum states of the atom and h is extremely small. One of the best-known alternative forms of the uncertainty principle is the time-energy uncertainty that can be written as:

${displaystyle Delta Ecdot Delta tau geq {frac {hbar }{2}}}$

This form is the one used in quantum mechanics to explore the consequences of the formation of virtual particles, used to study the intermediate states of an interaction. This form of the uncertainty principle is also used to study the concept of vacuum energy.

General expression of the indeterminacy relation

In addition to the two previous forms, there are other inequalities such as the one that affects the J_i components of the total angular momentum of a system:

${displaystyle Delta J_{i}Delta J_{j}geq {frac {hbar }{2}}left|leftlangle J_{k}rightrangle right|}$

Where i, j, k are distinct and J_i denotes the component of angular momentum along the x_i axis.

More generally if there are two physical quantities in a quantum system a and b represented by operators or observable denotated as ${displaystyle {hat {A}},{hat {B}}}$ in general it will not be possible to prepare a collection of systems all of them in the state ${displaystyle Psi ;}$ where the standard deviations of the measures a and b do not satisfy the condition:

${displaystyle Delta _{Psi }{hat {A}}cdot Delta _{Psi }{hat {B}}geq {frac {1}{2}}left|langle Psi |[{hat {A}},{hat {B}}]|Psi rangle right|}$

Demo

The general expression of the indeterminacy relation is deduced from postulates I and III of quantum mechanics. The more particular proof that there are magnitudes that cannot be known with arbitrary precision also makes critical use of the VI postulate.

To test Heisenberg's indetermination principle, let's suppose two observable. $scriptstyle A$ and $scriptstyle B$ any and suppose a state ${displaystyle scriptstyle |psi rangle }$ such as ${displaystyle scriptstyle {|psi rangleA|psi rangleB|psi rangle } subset D(A)cap D(B)}$ . In that situation it can be demonstrated that:

(1) ${displaystyle Delta _{psi }Acdot Delta _{psi }Bgeq {frac {1}{2}}|langle psi |[A,B]|psi rangle |}$

Where:

{displaystyle Delta _{psi }A={sqrt {langle A^{2}rangle _{psi }-langle Arangle _{psi }^{2}}}}

, the "incertidumbre" measured as standard deviation of the value of a measure on the state

|psi rangle

{displaystyle [A,B]=AB-BA,}

The switch of both observable.

Defining from $scriptstyle A$ and $scriptstyle B$ the self-adjunct operators:

${displaystyle {bar {A}}=A-langle Arangle _{psi },qquad {bar {B}}=B-langle Brangle _{psi }}$

You can build the actual function:

${displaystyle y(lambda)=langle psi |({bar {A}}-ilambda {bar {B}})({bar {A}}+ilambda {bar {B}})|psi rangle =|({bar {A}}+ilambda {bar {B}})psi |^{2}geq 0}$

And developing the previous scalar product:

(2) ${displaystyle y(lambda)=langle psi |{bar {B}}^{2}psi rangle lambda ^{2}+langle psi |i[{bar {A}},{bar {B}}]psi rangle lambda +langle psi |{bar {A}}^{2}psi rangle }$

Noting that:

${displaystyle [{bar {A}},{bar {B}}]=[A,B]}$
${displaystyle Delta _{psi }A^{2}=(langle psi |A^{2}psi rangle -langle psi |Apsi rangle ^{2})=(langle A^{2}rangle _{psi }-langle Arangle _{psi }^{2})}$
${displaystyle Delta _{psi }B^{2}=(langle psi |B^{2}psi rangle -langle psi |Bpsi rangle ^{2})=(langle B^{2}rangle _{psi }-langle Brangle _{psi }^{2})}$

The equation (2) can be rewritten as:

(3) ${displaystyle y(lambda)=(Delta _{psi }B)^{2}lambda ^{2}+langle psi |i[{bar {A}},{bar {B}}]psi rangle lambda +(Delta _{psi }A)^{2}}$

Like ${displaystyle scriptstyle [A,B]}$ is a hermitic operator the coefficients of the previous polynomial function are real, and as the previous expression is positive for all value of $scriptstyle lambda$ necessarily the discrimination of the associated polynomial must be negative:

(4) ${displaystyle langle psi |i[{A},{B}]|psi rangle ^{2}-4(Delta _{psi }A)^{2}(Delta _{psi }B)^{2}leq 0}$

Reordering and obtaining square roots in the previous equation is obtained precisely the equation (1). If the equation is particularized (1) taking ${displaystyle scriptstyle A=P, B=X}$ :

${displaystyle (Delta _{psi }X)(Delta _{psi }P)geq {frac {1}{2}}|langle psi left|[X,P]right|psi rangle |={frac {1}{2}}|langle psi left|ihbar right|psi rangle |={hbar over 2}}$

Estimating the energy of fundamental levels

Using the uncertainty principle it is possible to estimate the energy of the zero point of some systems. For this we will assume that in such systems the zero point fulfills that the particle would be classically at rest (at the quantum level it means that the expected value of the momentum is zero). This energy calculation method only gives an idea of the order of magnitude of the fundamental state, never being a method of calculating the exact value (in some system it may turn out that the value obtained is the exact one, but this is no more than a mere coincidence). The physical interpretation of the method is that due to the uncertainty principle, the location of the particle has an energetic cost (the kinetic energy term), so that the closer the particle is to the center of forces, the more energy the system will have due to to quantum fluctuations, so that at the fundamental level the system will minimize its total energy.

Particle in a Coulomb potential

Next, we will estimate the fundamental energy of a single-electron atom. By the uncertainty principle we have:

{displaystyle Delta rcdot Delta pgeq {frac {hbar }{2}}}

Using as an estimate that for the fundamental level is fulfilled:

{displaystyle Delta rcdot Delta papprox hbar qquad Rightarrow qquad Delta papprox {frac {hbar }{Delta r}}}

Total energy is the sum of kinetic plus potential. Since the mean value of the radial moment is zero, its expected squared value will be equal to its deviation and the expected value of the inverse of the radius will be approximated to the inverse of its deviation.

{displaystyle langle Erangle =langle Trangle +langle Vrangle =langle {frac {p^{2}}{2m_{e}}}rangle -langle {frac {1}{4pi epsilon _{0}}}{frac {Ze^{2}}{r}}rangle approx {frac {hbar ^{2}}{2m_{e}(Delta r)^{2}}}-{frac {1}{4pi epsilon _{0}}}{frac {Ze^{2}}{Delta r}}}

At the fundamental level the energy must be minimal so that:

{displaystyle {frac {dE}{dDelta r}}=0qquad Rightarrow qquad Delta r={frac {hbar ^{2}4pi epsilon _{0}}{m_{e}Ze^{2}}}=a_{0}}

The value obtained is coincidentally identical to the Bohr radius and substituting in the estimate obtained for the energy is obtained:

{displaystyle E=-{frac {(Ze^{2})^{2}m_{e}}{2hbar ^{2}(4pi epsilon _{0})^{2}}}=E_{0}}

Coincidentally this is exactly the ground state energy of a hydrogen atom. The objective of the method is the estimation of the value, although in this particular example obtained it is identical to the formally calculated one.

One-dimensional harmonic oscillator

Using as an estimate:

{displaystyle Delta xcdot Delta papprox hbar qquad Rightarrow qquad Delta papprox {frac {hbar }{Delta x}}}

Taking that the mean value of the position and moment are zero due to the symmetry of the problem, the total energy is:

{displaystyle langle Erangle =langle Trangle +langle Vrangle approx {frac {hbar ^{2}}{2m(Delta x)^{2}}}+{frac {1}{2}}momega ^{2}(Delta x)^{2}}

Minimizing energy:

{displaystyle {frac {dE}{dDelta x}}=0qquad Rightarrow qquad (Delta x)^{2}={frac {hbar }{momega }}}

Substituting the value in the energy we obtain:

{displaystyle E=hbar omega =2E_{0}}

As can be seen, the value obtained is twice the zero point of the harmonic oscillator, so that although the value obtained is not exact, the order of magnitude is correct.

Particle in a well

Let be a particle that is confined in an infinite well of width 2a. Since the only possible positions of the particle are inside the well, it can be estimated that:

{displaystyle Delta xcdot Delta papprox hbar qquad Delta xapprox aqquad Rightarrow qquad Delta papprox {frac {hbar }{a}}}

The kinetic energy will therefore be:

{displaystyle langle Erangle ={frac {langle p^{2}rangle }{2m}}approx {frac {hbar ^{2}}{2ma^{2}}}={frac {4}{pi ^{2}}}E_{1}}

As can be seen, the result obtained differs by a factor slightly greater than 2 from the real value, but again the order of magnitude is correct. This calculation gives an idea of the energies that must be provided to confine a certain particle in a region, such as a nucleon in the nucleus.

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