Max Born’s Statistical Interpretation | Quantum Science Philippines

## Max Born’s Statistical Interpretation

Liza Marie T. Dangkulos

This article contains a summary of Max Born’s Nobel lecture entitled, “The statistical interpretation of quantum mechanics”.

In 1926, shortly after the formulation of the Schrodinger’s equation, Max Born studied the scattering of a beam of electrons and was led to his interpretation of the wave function in the said equation.

Born’s statistical interpretation states that:
The probability of finding an electron, described by the wave function, Ψ (x,t), in the region lying between x and x+dx is given by:

where

is the complex square or Ψ*Ψ[1]

He, therefore, introduced the statistical point of view into modern physics.[2] For this invaluable contribution in the field of quantum mechanics, Born was awarded the Nobel Prize in Physics in 1954.

During his Nobel lecture, Born accounted the developments in the field of quantum mechanics that led him to his statistical interpretation. He mentioned that in 1925, he and Werner Heisenberg formulated the matrix mechanics representation of quantum mechanics. Wolfgang Pauli consequently calculated the stationary energy values of hydrogen atom by means of the matrix method and from this moment onwards, there could no longer be any doubt about the correctness of the theory.

In 1926, Louis de Broglie formulated the de Broglie hypothesis claiming that all matter, has a wave-like nature. He related wavelength (denoted by λ) and momentum (denoted by p) as:

λ=h/p

where h is the Planck’s constant

Schrodinger, following de Broglie’s wave-particle duality theory of matter, constructed his famous equation that describes how the quantum state of a physical system changes in time. This equation can be mathematically transformed into matrix mechanics.

Not long after, Born developed his statistical interpretation. Not only was it developed from Schrodinger’s equation but from Einstein’s idea as well. Einstein interpreted the square of the optical wave amplitudes as the probability density for the occurrence of photons. For Born, this concept could be carried over to the Ψ-function.  Ψ*Ψ represents the probability density for electrons.

Furthermore, Born also emphasized that the indeterministic statistical interpretation should be accepted despite the strong oppositions of some respected physicists like Erwin Schrodinger, Louis de Broglie and Albert Einstein. He believed that Heisenberg’s uncertainty principle contributed to the swift acceptance of the statistical interpretation of the Ψ-function.

Uncertainty principle states that certain pairs of physical properties cannot both be known to arbitrary precision. Its meaning, according to Heisenberg, is that it is impossible to determine simultaneously both the position and velocity of an electron or any other particle with any great degree of accuracy. With this, Born had this to say, “Can absolute prediction really be made for all the time on the basis of the classical equations of motion?”

Towards the end of his lecture, Born made these two statements: “Classical physics cannot be used as an objection to the essentially indeterministic statistical interpretation of quantum mechanics” and that “I am emphatically in favour of the retention of the particle idea.”[3]

Through his statistical interpretation, Max Born showed that the solution of the Schrodinger equation has a physical significance.

[1] Stephen Gasiorowics. Quantum Physics, 3rd ed. (John Wiley and Sons, Inc., 2003) p. 28.
[2] Walter Greiner. Quantum Mechanics: An Introduction, 4th ed. (Springer-Verlag, Berlin) p.65
[3] Max Born. The Statistical Interpretation of Quantum Mechanics. Nobel Lecture, 1954.