P. Jordan and V. Fock
At present in Kharkov
(Received 15 October 1930)
Zs. Phys.66, 206, 1930
If we assume that the most precise measurements of electromagnetic fields usingelectrons or protons as “test bodies” have been carried out, then by reason of the quantum mechanical uncertainty principle for the coordinate and momentum of the bodies, we get certain bounds on the measurabilityof the field intensities, namely, the same bounds for mea- surements both with electrons and protons. In the sense of the quantum mechanical concepts, it means that these fundamental restrictions on themeasurability of the electromagnetic fields suggest a reasonable lim- itation on the possibilityto determineprecisely the state of the classical field.
Electric Field Intensities
In order to measure an electrical field, varying in space and time, at a pointx, y, z at time t, we place an electron which we represent as a wave packet localized at the pointx, y, zwith zero velocity1at this time t. Then we determine the corresponding acceleration of the electron caused by the field. Being averaged over a small time-intervalt, t+δt for thex-componentEx it is given by the equation
−eEx=mδvx
δt
(−eandmstand for the charge and mass of the electron, respectively), whereδvx is the increment of the velocity in thex-direction of the elec-
1In order to exclude a simultaneous deflection of the electron due to the magnetic field. (Authors)
tron within this time interval. Now we can attribute a certain velocity componentvxto timetand to timet+δtonly with a finite accuracy ∆vx, which is connected with the uncertainty of the electronx-coordinate
m∆vx∆x≥~.
Consequently, the measured value ofEx acquires an indefiniteness that has the following estimate:
∆Ex=m∆vx
eδt ≥ ~ e
1
∆x∆t.
Hence, we can measure only an average of Ex, which refers to a space interval of the length ∆xand to a time interval of the duration
∆t=δt. The uncertainty ∆Ex of the valueEx is inversely proportional to the product ∆x∆tof the corresponding valuesxandt, on which the measured quantity ofEdepends.
As a complementation of the corresponding equations forExandEy, we obtain
∆Ex∆x∆t ≥ ~ e,
∆Ey∆y∆t ≥ ~ e,
∆Ex∆z∆t ≥ ~ e.
(1)
Magnetic Field Intensities
The same considerations are also valid for magnetic field intensities. Be- low we investigate two cases that are distinguished by the curvature of the electron trajectory.
a) Trajectories of sharp curvature. Consider an electron having a helical trajectory in a magnetic field which is directed along thex-axis.
Then we obtain
Hx= mv a
c e = p
a c e,
whereais the radius of the orbit. The uncertainty ofHx is
∆Hx=∆p a
c e.
Now a is the uncertainty of the coordinate in the direction of the radius and ∆p, of the momentum in the perpendicular direction; we can set, e.g.,
a= ∆r; ∆p= ∆py. Owing to
∆py≥ ~
∆y, we obtain
Hx≥~c e
1
∆y∆z.
b) Trajectories of small curvature. We denote byvIthe unit vector in the direction of electron velocity and byH⊥the component ofH, which is perpendicular tovI. Then we obtain
H⊥= c
evI×mδv δs .
We suppose that the electron moves in they-direction, therefore, vIx= 0, vIy= 1, vIz= 0, δs=δy
and thex-component ofHis Hx=c
e mδvz
δy = c e
δpz
δy.
We hold the value ofy and measure the corresponding shift δpz of the momentumpz. By the order of magnitude, the uncertaintyδpz is equal to ∆pz, which gives for the uncertainty ofHxthe relation
∆Hx= c e
∆pz
δy .
Let furtherδybe the uncertainty of they-value, to which the measured value ofHx corresponds, i.e., δy = ∆y. Because of the relation ∆pz ≥
~
∆z, we obtain, as before,
∆Hx≥~c e
1
∆y∆z.
This relation and those that are obtained from it by the cyclic permu- tations of indices are written in the following form:
∆Hx∆y∆z≥~c e,
∆Hy∆z∆x≥~c e,
∆Hz∆x∆y≥~c e.
(2)
Another Form of Relations (1) and (2)
Obviously, equations (1), (2) have the correct relativistic symmetry. It is also possible to formulate their contents as follows: let us consider an in- tegral over an arbitrary two-dimensional surface in the four-dimensional coordinate–time space
J = Z
FàνdSàν,
whereFàν are the electromagnetic field intensity components anddSàν is an infinitely small two-dimensional surface element. Thus,J is defined with the uncertainty2
J= ~c e.
The same reasoning is also valid for the integral J =
I
Φàdxà,
which is taken along the closed contour in the coordinate–time space where Φà denotes a 4-potential.
Concluding Remarks
The contemporary quantum theory of electromagnetic fields fails to give well-justified deductive derivation of equations (1) and (2). It states that a possibility to measure any field component, e.g.,Ex, with an arbitrary precision at an exactly defined space–time pointx, y, z, t is excluded by
2We were informed about this formulation by D. Ivanenko. See also J.Q. Stewart, Phys. Rev.34, 1290, 1929. (Authors)
our relations. Furthermore, according to Heisenberg,3the theory permits a simultaneous measurement of the field intensitiesExandHywithin the space volume (∆l)3 with the uncertainty
∆Ex∆Hy≥ ~c (∆l)4,
while, conversely, our theory gives the uncertainty for separately mea- sured intensities of Ex and Hy the value that is obtained by the multi- plication of the quantities mentioned above,
∆Ex∆Hy ≥ ~2c2 e2(∆I)4,
with the minimal value, which is greater by a factor ofhce2. The derivation of equations (1) and (2) on the basis of a proper multiplication law for field operators representing field quantities is expected to be given only by a future theory that calculates quantities with the accuracy of the fine structure constant.
Kharkov,
Physical-Technical Institute Translated by A.V. Tulub
3W. Heisenberg, The Physical Principles of Quantum Theory, Leipzig 1930.
30-5