Quantum Mechanics and Emergence of Quantum Theory: Analytical Essay

Introduction

Until the end of the 19th century, physics was able to expalin the mechanic, acoustic and thermal phenomena by Newton’s Laws and electric, magnetic and optics by Maxwell’s Equations at the macroscopic scale. Even though these two great phenomena showed separation between wave-like behavior and corpuscular character of matter, some observations could not be explained by classical physics while some of them, like emission, radiation etc., disproved the separation between particle and field. Physics had to be rebuilt, leading to the emergence of quantum theory.

The first problem to be solved was that of blackbody radiation. In the beginning of 20th century, Wien’s law does not apply to infrared light is shown by Heinrich Rubens and Ferdinand Kurlbaum which indicates that the peak wavelength of a blackbody only depends on its temperature. In 1901, Max Planck described the behavior of some types of radiation by assuming that objects absorb and emit light that has an energy proportional to the light’s frequency and he showed that the energy exchange between matter and radiation is not continuous but discrete which is called as quantizing that is only certain energies are allowed. So the energy quantization was the first step toward a quantum theory able to give reason for the experimental data, incompatible with the classical theory.-

Philipp Lenard proved that the energies of the electrons does not depend on the intensity of light, but frequency in 1902. In addition to that, Einstein showed that released electrons from matter when it absorbs ultraviolet light, are only released when particular frequencies are reached that are multiples of Planck’s constant which could not be explained using Maxwell’s theory of light.

Thomas Young showed that the light consisted of waves at his very known experiment, double slit, in the beginning of 19th century and disprove the corpuscular theory of light proposed by Isaac Newton. Photons come together and perform an interference pattern right after splitting while going through the slits.

This created theory demonstrates a wave-particle nature of elementary particles, confirms the existence of not continuous but discrete energy levels as claimed by Planck, and provides a lower limit for how exact a measurement can be by Heisenberg’s uncertainty principle and defines wave function, ψ, that is the solution of the Shrödinger equation and gives all the information of a quantum mechanical state for which is a complete description.

The completeness of the quantum mechanics is argued in the paper titled “Can Quantum-Mechanical Description of Physical Reality Be Considered Complete?” written by Albert Einstein, Boris Podolsky and Nathan Rosen in 1935. They showed that quantum mechanics cannot satisfy locality, reality and completeness at the same time via a gedankenexperiment later named as EPR Paradox. As a result, either quantum mechanics was incomplete or there exists some hidden variables that determine the state of the particles all along.

In 1964 John Steward Bell published a paper titled “On the Einstein Podolsky Rosen Paradox” which he came up with inequalities considering local theories with hidden variables and showed that quantum mechanics violate those inequalities which mean hidden variables do not exists.

EPR Paradox

Introducing the Problem

In 1935 Einstein, Podolsky and Rosen published a paper titled ”Can quantum-mechanical description of physical reality be considered complete?” which they discuss that although quantum mechanics is a very successful description of nature it cannot represent a final complete theory. The particular matter that led them to this conclusion was a puzzle that comes around when the states of two particles are entangled that is some strong correlations between particles. The state of a particle cannot be known until a measurement is done in quantum mechanics. In such an entangled state, characterization of the states are given by a wave function which the individual particles are in ambiguous state until one of them is measured.

If a measurement is made on one of the particles when the particles are so far away from each other that they do not interact to one another with any known classical potential, the state of the other particle became instantaneously well-defined which is collapse of the wave function regardless of the distance of the two parts. That is a non-local behavior that the theory assumed to claim and called by Einstein as “spooky action at a distance” which is a violation of one of Einstein’s Special Relativity postulates that is no information can travel faster than the speed of light because this would violate causality. They gave the following working definition “a sufficient condition for the reality of a physical quantity is the possibility of predicting it with certainty without disturbing the system”, then presented a thought experiment designed to show that quantum mechanics cannot be a complete description of reality.

Define Reality, Locality, Completeness

A physical reality is determined by outcomes of measurements or experimental results. As Einstein et al. stated in the paper “if a value of a physical quantity can be predicted with certainty then there exists a physical reality corresponding to this physical quantity that is independent from external observers.”

Since the velocity is upper limited by the speed of light,c, by the special theory of relativity, the principle of locality signifies that there can be no simultaneous effects from one point to another. In a different manner, the effect cannot happen less than in t = D/c where D is the distance between two events and c is the speed of light.

If a theory consists of elements of physical reality has no counterpart in the theory then it is said that the theory is not complete.

Experiment

Thought Experiment

Suppose at the origin of the x-axis some event emits two identical particles which one travels to the right with momentum and position and the other to the left with momentum and position . If the total momentum of the system was initially zero, then to conserve zero momentum would have to be the opposite of so that their sum is zero. Because the particles are identical, opposite momentums means opposite velocities, so position would at all times be the negative of position as it shown on the Figure 1. Quantum mechanics says that the position and momentum of one particle cannot both be real, according to the EPR definition because of the uncertainty principle. If one shrinks the one uncertainty to zero either or , then the other uncertainty has to grow to infinity so that their product is never less than Planck's constant according to quantum mechanics thus if the position of a particle is real then it's momentum cannot be real and vice-versa.

Figure 2.1. Thought Experiment

If position measurement is done on any particle, say #1, and result is say , then it is also known that due to the relationship between the positions, so one can predict with certainty without having disturbed particle #2 so becomes real and instead if measurement is done on the momentum then we can predict with certainty again without having disturbed particle #2 so becomes real. In this experiment, reality for particle #2 depends on measurements made on particle #1 and these particles can be arbitrarily far apart that violates principle of locality.

EPR Formalism

Physical quantity A has a definite value of when it is in the state described by the wave function where If is an eigenfunction of the operator A where is real,

(2.1)

If a particle in a state given by the wave function where the Eq. (2.1) is satisfied then there should be an element of reality corresponding tho the physical quantity A by the definition of reality. Where h is the Planck’s constant and is a constant.

(2.2)

With using the definition of momentum operator,

(2.3)

One obtains,

(2.4)

Which means that momentum has definite value at the state given by Eq. (2.2) therefore the momentum of the particle in the very state is real, by the definition of reality.

If the Eq. (2.1) is not valid which means the quantity A has no certain value, say position of a particle, the operator that corresponds to it, say q, is the operator of multiplication by the x, the independent variable. So,

(2.5)

The relative probaility of a position measurement gives the result between the boundaries of a and b according to quantum mechanics,

(2.6)

The result is not only depends on a but the difference of b-a, therefore all values of the coordinate are equally probable.

In other words, the value of the position of the particle given by the state Eq. (2.2) is not certain therefore it can be only be determined by direct measurement which disturbs the particle and changes its state. Thus when the particle’s momentum is known then the position has no physical reality. As it is shown in the Heisenberg’s uncertainty principle, if two physical quantities do not commute then both values cannot be known precisely.

The description of reality given above contrasts with the wave function which is supposed to include complete description of the reality of a quantum mechanical system. To Show this contradiction, a two particle system is assumed that are in interaction between t=0 to t=T then there are no interactions. By using the Schrödinger Equation the state of the system can be determined for any time.

Let the a physical quantity, say A relative to the first system, has eigenvalues as and the eigenfunctions . Then the corresponding wave function can be written as follows where describes the variables used to describe first system, describes second, are the coefficients of the expansion of the wave function by the series of orthogonal functions i.e. Fourier’s coefficients .

(2.7)

If one measures the A and finds the value as then the first system collapses to the state given by , that is,

(2.8)

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Place Order

After the measurement is done on the first system then the second system will collapse into a definite state . The process is called as reduction of the wave packet which from the series given by Eq (2.7) it is reduced to single term given by Eq (2.8).

Then define a new quantity that is B with eigenvalues as and the eigenfunctions and with help of the equation (2.7) where are the new coefficients.

(2.9)

If one measures the B and finds the value as then the first system collapses to the state given by and the second system collapses to the state given by .

As a result, second system is left in states given by two different wave functions by two different measurements made on the first system while two systems do not interact. Therefore two different wave functions possibly be assigned to one reality.

Let and are physical quanties that has eigenfunctions of and respectively and assume both of the system’s are particles then the total wave function of the system is where is a constant,

(2.9)

Let the physical quantity A be the linear momentum P which Eq (2.3) shows the operator form of it. Measurement of the quantity at first system will give a value p therefore the particle will fall into a state where the corresponding eigenfunction of it will be as it is in Eq (2.4),

(2.11)

By using Eq (2.11) and considering a continuous spectrum the wave function of the total system,

(2.12)

Where,

(2.13)

This wave function is nothing but eigenfunction of the opertator P which has eigenvalue –p of the second particle.

(2.14)

Moreover, let B donate the position of the first particle that has eigenfunctions of the eigenvalue x are

(2.15)

Where the right hand site of the equation are Dirac delta function. The total system’s wave function then becomes by using Eq. (2.8)

(2.16)

Where,

(2.17)

That is the eigenfunction of the operator Q with corresponding eigenvalue of the second particle.

(2.18)

The commutator of both operators is

(2.19)

Thus, two non-commuting operators, P and Q, have the wave functions and of the second system that can be simultaneously real.

Bohm-Aharonov Interpretation

Einstein, Podolski and Rosen used the variables which are position and momentum to characterize the problem yet it was hard to observe the problem at those entangled states, because the eigenfunctions for these operators were delta functions, δ(r − r0), and their Fourier components, the exponentials . David Bohm adapted the same problem to a pair of spin one-half particles which are entangled. At the very state, the system can be defined as the wave function given by the following equation where and are the positions of the particle 1 and particle 2 respectively and the suffixes are the spin states.

(2.20)

Spin components of both particles are ill-defined. If one measures the spin in the z-direction of any particle, its state immediately become well defined with value of or and that measurement also makes other particle’s spin state well defined instantaneously but opposite of the first no matter how far apart they are.

Only this argument is not dissimilar than picking gloves from two boxes because if the one opens any of the boxes and finds that it is the left one then it is directly known what’s inside the other box but the case is different at the quantum state than the classical state therefore the same wave function can also be written as along the x-axis as

(2.21)

As it is done at the z-axis if one of the particle’s x-component of spin is measured which would yield either or implying other particle is in the opposite state with respect to the measured particle. These two individual cases are possible however quantum mechanics states that a particle cannot be both ψ±z and ψ±x at the same time because their commutator is non-zero.

(2.22)

Result of EPR

EPR showed that two non-commuting quantities which are P and Q can have simultaneous reality by starting from the assumption that the wave function is a complete description. Einstein et al. stated that in such a system that non-commuting operators of a two-physical quantities, the information of one prohibits the information of the other. If so, (1) reality definition that comes from the wave function is not complete in the quantum mechanics or (2) these two quantities cannot have simultaneous reality. If (1) is false then the (2) is also false where the conclusion of a problem that is making predictions about one system based on the measurements of the another system which are both initially interacted. One is thus led to conclude that the description of reality as given by a wave function is not complete. This procedure makes the reality of P and Q depending on

the measuring process that is executed on the first system, which does not interfere in any way with the second one that has contradiction with the definition of the locality and in EPR words “spooky action at a distance”. As EPR concluded, “No reasonable definition of reality could be expected to permit this.”

Einstein postulated the existence of hidden variables: as yet unknown local properties of the system which should account for the discrepancy, so that no instantaneous spooky action would be necessary. EPR paradox forces to conclude that either quantum mechanics is incomplete or the concept of reality needs a radical revision.