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NeuroQuantology | March 2010 | Vol 8 | Issue 1 | Page 101‐109 Erol M., Schrödinger wave equation and function, relations with consciousness

ISSN 1303 5150 www.neuroquantology.com

101

Basics of Quantum Physics

Schrödinger Wave Equation and Function: Basics and Concise Relations with

Consciousness/Mind

Mustafa Erol Abstract Scientists used to believe until recently that the consciousness/mind/thought is apurely metaphysical concept and hence has nothing to do with science. However,recent scientific research both theoretically and experimentally strongly imply that consciousness/mind/thought is in fact a physical concept and could be investigatedby scientific means. This very exciting and revolutionary approach recentlyattracted much interest to the subject. However, the nature of the subject needsinterdisciplinary collaboration of especially neuroscientists and physicists/quantumphysicists and also possibly psychiatrists, philosophers and more. This paper istherefore organized and prepared in order to serve to bridge the gap betweenespecially neuroscientists and quantum physicists. The present work in particular deals with the Schrödinger wave equation and wave function that lie right on thespine of the entire quantum physics and must be understood clearly before anyfurther steps. More specifically, the paper focuses on philosophical approaches to consciousness, Schrödinger wave equation, wave function and its physical meaning,generalization of the wave function and finally general remarks on the relationbetween consciousness/mind and quantum physics. Scientific and philosophical background is momentarily fulfilled in advance to enhance the emotional willingness of the readers. Key Words: quantum physics, Schrödinger wave equation, wave function, consciousness, mind, brain

NeuroQuantology 2010; 1: 101‐109

1. Introduction1 Scientific research on relation between consciousness/mind or brain and quantum mechanics has not been progressed sufficiently even though the roots go down to 1925’s (Lotka, 1925; Bohr, 1928, Neumann,

Corresponding author: Mustafa Erol, Prof.,

Address: Department of Physics Education, Education Faculty of Buca, Dokuz Eylül University, Buca, İzmir, 35160, TURKEY

Phone: + 232 4204882/1311 Fax: + 232 4204895 e‐mail: [email protected] Received Jan 18, 2010. Revised Feb 15, 2010. Accepted March 17, 2010.

1932; Whitehead, 1933; Bohm, 1952; Eccles, 1973; Walker, 1970; Bass, 1975). One of the founders of the quantum physics, Bohr himself, underlined the possible link between quantum mechanics and human nervous system/consciousness, however even very fundamental topics are still left to be unresolved. Its now excitingly promising that the scientific research has recently been increased dramatically, especially following the apparent interest of the quantum physicists to the subject and recent research giving very strong positive signals about the



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sparkle future of the subject (Beck, 2008; Conte, 2008; Stapp, 1991; Hameroff & Penrose, 1996; Vitiello, 2003; Jibu & Yasue, 1995; Stapp, 1993, 1995; Khrennikov, 2005, 2006a, 2006b; Tegmark, 2000; Penrose, 1994; Conte et al.,, 2008, 2009a, 2009b; Eccles, 1994; Beck&Eccles, 1992). The previous research has not been able to resolve the mind/consciousness due mainly to that neuroscientists tried to resolve the mind/consciousness and also brain within the discipline by purely considering the neuroscientific/biological principles and also some classical principles of physics. However, it is a solid reality that any matter, including the living matter hence the brain, must be comprised of tiny particles at nano scales such as molecules, atoms and sub atomic particles, and all those particles are governed purely by quantum mechanical laws and needed to be treated accordingly.

The other point is that the subject seems to be too outsized to just handle with purely scientific view so, to our view, it also needs some philosophical approach. Consciousness/mind is considered a metaphysical concept through the centuries both scientifically and philosophically. Philosophically approaching to the subject concludes well known anthropic mechanism and it doctrines that “the behavior of living matter can completely be understood by purely physical laws”. Human mind/consciousness is actually disqualified in that approach. However, recent scientific work very strongly suggests that human mind/consciousness can also be fully explained by the physical laws. This approach can be considered as extended anthropic mechanism. Based on this approach, we hypothesize the following by considering the present scientific research and adding a fragment of philosophy. Hypothesize 1. Consciousness/mind/thought is a pure physical concept and energy, establishes at a time level of about 0.1s, space level of about 10-15 m and energy level of about 10-15 eV. Therefore it is well in the quantum regime and must be treated accordingly. Hypothesize 2. Brain and mind/consciousness/thought are identical and no separable (same) concepts at that energy and space levels and there is no “binding problem” as such.

One of the most apparent outcomes of the quantum theory is the discreteness of the energy and some other physical concepts. Hypothesize 1 emphasizes that the consciousness/mind is essentially a physical energy so must be quantized and there must exists “consciousness/thought quanta”, previously named as “psychon” by Eccles, which could more conveniently be named as “thoughton” or may be abbreviated as “Ton” (Eccles, 1986; De and Pal, 2005). Therefore, the readers are supposed to study the text by simply substituting “particle/atom/sub atomic article” or “quantum particle” by “thought quanta-Ton”.

The rest of the paper briefly underlines the fundamental laws and principles concerning “Schrödinger Wave Equation” and “Wave Function” which have a central role in quantum mechanics in order to back the needs of especially neuroscientists.

2. Schrödinger Wave Equation (Eisberg and Resnick, 1970) Quantum physics is the theoretical framework with in which it has been found possible to describe, correlate and predict the behavior of a vast range of physical systems from molecules, atoms down to subatomic particles. Matter at the atomic and subatomic levels reveals the existence of a variety of particles which are identifiable by their distinct properties such as energy, charge, spin and magnetic moment all of these seem to be of a quantum nature in the sense that they take on the certain discrete values this discreteness of physical properties persists when sub atomic particles form atoms or molecules. At atomic and subatomic levels, physical properties of any particle is determined by a well known second order differential equation that is essentially confirming that any particle with a mass of m and total mechanical energy of E is accompanied by a wave function, ψ(x, y, z, t) where x, y and z denote physical positions of the particle at normal space and t is the time. This wave equation is, in terms of functioning, similar to the Newton’s Laws of classical mechanics and has the central significance in Quantum Physics. This wave equation is called Schrödinger Wave Equation (SWE) and any atomic or sub



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atomic particle must obey to this equation and any behavior of the particles has been determined by this universal equation. SWE can be established in simplest way by considering “Eigenvalue - Eigenfunction” equation that is mathematically well defined for any operator. Operators are general concepts that transform a mathematical function to a different one. The equation is mathematically formulated as,

=A. f ( x ) a. f ( x ) (1)

where A denotes the operator, f(x) is a one-dimensional mathematical function known

as eigenfunction of the operator A and a is

the eigenvalue of the operator A .

This elegant equation has very deep influence on the physical universe since in quantum physics any concept such as position (x), time (t), momentum (p), kinetic energy (K), potential energy (V) or total mechanic energy (E) is represented by an “operator” hence all those concepts can be applied and must be satisfied the equation (1). We also experimentally observed nearly a century ago “diffraction of electrons” from solid state crystals clearly demonstrating that electrons are accompanied by waves thus their behavior must also be described by a wave function and wave equation. We now want to discover this wave equation, namely Schrödinger Wave Equation, as simplest as possible by considering the arguments mentioned above. Total mechanical energy of any particle which is also classically known as “Hamiltonian” is given by the addition of kinetic energy (K) and potential energy (V). Hence, considering only one dimension that is x, and time that is t the Hamiltonian of a particle is given by,

H(x,t)=K(x,t)+V(x,t) (2)

The equation above assumes that all concepts are time and also position dependant. It is very important to underline at this point that the systems or particles in the physical world is generally isolated from the environment and the total energy is conserved or the Hamiltonian is generally independent of the time. The living matter, such as brain or mind/consciousness, on the

other hand, continuously interacts with the environment and consequently the energy sum for the “consciousness quanta” can not be considered as constant. Note that in the text below, the total energy of the quantum particle is primarily considered to be conserved and treated accordingly. The simplest way to derive SWE is to define the “Eigenvalue – Eigenfunction” equation given in the equation (1) for the Hamiltonian, that is

=H( x,t ) ( x,t ) E ( x,t )ψ ψ (3)

where Ĥ(x,t) is the position and time dependent Hamiltonian operator, E denotes the total energy obtained by the measurement of Ĥ and finally ( x,t )ψ is the mathematical expression of the wave accompanying the particle at point x and time t (Schrödinger, 1935). We now need to establish the Hamiltonian operator for a quantum particle. As we aim to launch a wave equation and/or wave function for the quantum particles, it is very convenient to start with the position and time dependent classical wave function expression that is

−= i( kx t )( x,t ) A.e ωψ (4)

where A denotes the amplitude of the wave, e=2.718 is the base of natural logarithm,

1= −i is the imaginary unity, 2= fω π is

the angular frequency, 2

=kπλ

denotes the

wave number, f denotes frequency and finally λ is the wave length. If we take the first derivative of ψ (x,t) with respect to time t, we acquire

∂= −

∂( x,t )

i ( x,t )t

ψ ωψ (5)

and take the first derivative of ψ (x,t) with respect to position x, we catch

∂=

∂( x,t )

ik ( x,t )x

ψ ψ (6)



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where ∂∂t

denotes partial derivative with

respect to t and ∂∂x

denotes the partial

derivative with respect to x. Before starting to establish the actual SWE, it is very advantageous to mention the principle of Planck which is “energy quantization” and principle of Broglie which is “wave-particle duality” both observed and proved solidly for any phenomena especially at atomic or sub atomic levels. Planck’s Energy Quantization: Planck formulized that any particle/matter or radiation/light at atomic or sub atomic levels must have a total energy of E=nhf where n is a positive integer (n=1,2,3,4….∞), f denotes linear frequency and h is a universal constant known as Planck’s constant, numerically is equal to h=6,626.10-34J.s. The energy quanta is then given by

E=hf= ω where 341 054 102

−= =h

, . J.s.π

and known as reduced Planck’s constant (Planck, 1901).

Broglie’s Wave Particle Duality: Broglie proposed that any radiation/light or particle/matter moving with a momentum of p accompanied by a wave length of λ, is related to each other by the equation of

=h

pλ

where momentum is a elementary property of particles and λ is a universal property of any wave. The expression above converted to a different version by simply dividing by 2π

which gives =p

k (Broglie, 1924). We now

can obtain time dependent total energy operator that is time dependant Hamiltonian operator from the equation (5) as

∂ ∂= − =

∂ ∂H(t ) i

i t t (7)

and by using the equation (6), the position dependent momentum operator can be found as

∂ ∂= = −

∂ ∂p( x ) i

i x x (8)

In order to get the position dependent total energy operator that is Ĥ(x), we ought to substitute the momentum operator (8) in the general expression of the Hamiltonian given by (2). It is important to note here that the potential energy of the quantum particles is considered to be only position dependant as is true for almost all the physical systems under consideration. However, similar to the total energy argument above the quanta of the brain or consciousness/mind, “ton”, is continuously environment influenced and can not be considered as a time independent concept. Under the illumination of the arguments above, the position dependant Hamiltonian operator can be found as (Schrödinger, 1935)

2

2 2

2

2

2

= + =

∂+ = − +

∂

p ( x )ˆ ˆ ˆH( x ) K( x ) V ( x )m

ˆ ˆV ( x ) V ( x )m x

(9)

The above equation assumes that the potential energy for the quantum particle is only position dependant; however, it should be mentioned that in the case of consciousness/mind it is likely to be time dependant due to being interacting constantly with the environment (Conte, 2009b).

3. Time Dependent Schrödinger Wave Equation Time dependent SWE can, at this moment, be obtained by substituting time dependent Hamiltonian operator given by (7), in the equation (3)

∂= =

∂(t )

H(t ) (t ) i E (t )t

ψψ ψ (10)

The wave function in the equation above is expressed as only time dependant, because the Hamiltonian is considered to be only time dependant. This equation is commonly known as the “time dependant SWE”.



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Concerning the total energy of the quantum particle, E, it is very important to separate two distinct situations: firstly, E can be considered constant (time independent) as most of the physical systems and particles convey. The solution of the time dependent SWE for a particle with a constant total energy of E=E0 is quite straightforward and given by

0

0

−=

Ei t

(t ) .eψ ψ (11)

where 0ψ is the initial value of the wave

function at t=0, known as the phase constant, and

00 02= =

Efω π

denotes the angular frequency of the wave for a constant energy of E0. This expression has a very special meaning of being universal and invariant; hence, this term is just standard for any quantum mechanical problem or state. Secondly, considering specifically the quanta of the consciousness, it is quite clear that as it continuously interacts with the environment, its total energy must be time dependant, in other words, energy must be mathematically a function of time, E=E(t). In this case the solution for the equation (10) can be expressed by

0

00

− ∫=

tiE E ( t )dt

i t(t ) .e .eψ ψ (12)

where E(t) denotes time dependence of the total energy. Time evolution of any quantum mechanical particle is determined by this equation. The equation is especially vital for mental structures because just about all the processes concerning brain or consciousness/mind are temporally distributed and therefore should be obeying the simple equation above (Khrennikov, 2006b).

4. Time Independent One Dimensional Schrödinger Wave Equation Similarly, substitution of position dependent Hamiltonian operator given by (9) leads to the position dependent SWE. Hence, a quantum mechanical particle with a mass of m, potential energy of V(x) and total energy of E accompanied by the wave function

( x )ψ must satisfy the time independent wave equation given below (Schrödinger, 1935).

2 2

22

∂= −

∂+ =

( x )H( x ) ( x )

m xV ( x ) ( x ) E ( x )

ψψ

ψ ψ (13)

where the total energy E=E0 is assumed to be constant (position independent) as holds for most standard physical systems. The wave function here is thought to be only position dependant due to be employing only position dependant Hamiltonian. If the quantum mechanical particle with a mass of m, potential energy of V(x) and total energy of E moves in one dimension, then both time and position dependant wave function , ( x,t )ψ , can be written as the product of the time dependant and position dependant wave functions, that is

=( x,t ) ( x ) (t )ψ ψ ψ (14)

where ( x )ψ is the only position dependent wave function obtained from the equation (13) and (t )ψ is the only time dependent wave function given by (11) or (12). The equation is strongly dependent on the potential energy and total energy of the particle. So any particle having different potential energy would conclude different solutions. Rearranging the time independent one dimensional SWE and considering E=E0, gives the most commonly known version of it, that is

2

02 2

20+ − =

d ( x ) m[ E V ( x )] ( x )

dx

ψ ψ (15)



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It ought to be underlined that behavior of systems or particles having position dependant total energy, E=E(x), can be simply analyzed by substituting E0 by E(x). It is strongly believed that all the processes concerning consciousness/mind or brain is also determined by the spatial distribution of the wave function and probability density which is determined by this equation. However, it should also be noted that unlike most of the physical systems under consideration so far, the total energy for consciousness/mind quanta, is likely to be position dependant and must be treated in the view of that.

In generalization of this work, we see that if the particle is confined spatially then Schrödinger’s equation yields a set of solutions n ( x )ψ which are complete in the

sense that any specific energy of En with a corresponding n ( x,t )ψ must also satisfy the

original form of the wave equation that is Eigenvalue-Eigenfunction equation,

=n n nH ( x,t ) E ( x,t )ψ ψ (16)

where n indicates the relevant quantum state and is known as principle quantum number. According to this equation, any quantum mechanical particle with a mass of m, total energy of E and potential energy of V(x) would physically have possible total energies of En, specifically, E1, E2, E3, ….., En. The corresponding wave functions accompanying the particle can be expressed as

1 2 3 n( x,t ), ( x,t ), ( x,t ),...., ( x,t )ψ ψ ψ ψ . Each

of the wave function n ( x,t )ψ is a particular

solution of the Schrödinger’s equation for the same potential energy of V(x) and is said to be a member of mathematically well defined “Hilbert Space” that is expressed by,

H={ 1 2 3 n(x,t), (x,t), (x,t),...., (x,t),...ψ ψ ψ ψ } (17)

The Hilbert Space is comprised of Eigen functions of the Hamiltonian operator, which is a hermitic, commute and linear operator, as all the operators used for physical concepts must be. The Hilbert space therefore is an orthonormal space which means it satisfies both orthogonally and normalization condition. This property can

mathematically be expressed by Kroenecker delta function that is

1

0

∞

−∞

= ⇒⎧ ⎫= = ⎨ ⎬≠ ⇒⎩ ⎭∫ *

mn m n

m n( x,t ). ( x,t ).dx

m nδ ψ ψ (18)

where m and n denotes different quantum states or energy levels. Since the wave equation and the wave function is linear then we expect that any linear combination of these functions will also be a solution of the equation. This can clearly be extended to show that an arbitrary linear combination of all wave functions which are solutions to the Schrödinger’s wave equation (SWE) can be given as

1 1 2 2 3 3

1

∞

=

= + +

+ + + =∑n n n nn

( x,t ) c ( x,t ) c ( x,t ) c ( x,t )

... c ( x,t ) ... c ( x,t )

ψ ψ ψ ψ

ψ ψ(19)

This general principle is called “superposition principle”. In fact, this expression, ( x,t )ψ , gives the most general form of the solution to the Schrödinger’s equation for a potential energy of V(x). Its generality can be appreciated by noting that it is a function, which is composed of a very large number of different functions combined in proportions governed by the adjustable constants, cn.

5. Physical Meaning of the Wave Function We at this instant want to discover what the wave function of ψ(x,t) physically means? The wave function ψ(x,t) has no physical meaning on its own other than being a mathematical expression however the relation between the wave function and the behavior of the associated particle must be described. The quantum particle, associated with the wave, must be at a specific time t and location x where the waves are spread in space and can only have an amplitude and phase at that point and time. Talking about the physical meaning of the wave function, Born’s interpretation also known as Copenhagen interpretation, states that the measurable physical quantity in this case is “probability density-P(x,t)” and is defined as the probability per unit length of x axis, of



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finding the particle near the coordinate x and time t. The finding probability must be real and non-negative, whereas the wave function ψ(x,t) is complex and obviously not possible to equate P(x,t). However ψ*(x,t).ψ(x,t) is always real and non-negative and can be correlated to probability density by P(x,t)= ψ*(x,t).ψ(x,t). The wave function and so the probability density P(x,t) must be continuous in space then the probability P(x,t).dx is defined as the finding probability of the particle within dx , in other words, between x and x+dx. The overall probability can then be found by integrating the expression for the all space that is

Total ∞

−∞

= = ∫ *Pr obability P ( x,t ). ( x,t ).dxψ ψ

(20)

It is well known fact that the total finding probability of any particle within all allowed space must be equal to unity so

Total 1∞

−∞

= = =∫ *Probability P (x,t). (x,t).dxψ ψ

(21)

can easily be written. In order to justify the expression above, we consider the classical definition of “mean value” given for any variable. If the variable of position is given for the state of n by xn and the probability of being xn is given by Pn(x), then the mean value is classically defined by Mean value

=1=

= = =∑ ∫f

i

xN

nn nx

x x P x P( x )xdx (22)

In the equation above the integral form is to be employed in the case of continuous variables and in that case instead of Σ, integration form must be used. The situation for the quantum mechanical particle is just analogous to the classical situation. The mean value of the position for a quantum mechanical particle is called “expectation value” and by using the equation (22), it can mathematically be defined by

+∞

−∞

= = ∫ *x x ( x,t )x ( x,t )xdxψ ψ (23)

The description given above can be generalized to any quantity or concept such as energy E or momentum p.

We now want to determine the probability of Pn that is finding the quantum particle at a the quantum state of n and energy of En. In order to do so, we substitute the expression (19) into the equation of (21) and also employ the equation (18) then we obtain,

∞

−∞

= = ∫ *Pr obability P ( x,t ). ( x,t )dxψ ψ =

2

1 1

1∞ ∞

= =

= =∑ ∑n nn n

c P (24)

The equation shows that the constant cn has a very important meaning of probability.

Specifically Pn=2

nc means total existence

probability of the particle at the quantum

state of n and energy of En. Since Pn=2

nc

physically means finding probability then the sum of all the probabilities must be equal to unity. The definition of the constant cn can easily be formulated by substituting the equation (19) within the equation (20) and using again (18),

22

∞

−∞

= = ∫ *n n nP c ( x,t ). ( x,t )dxψ ψ (25)

which indicates the importance of the interaction between any specific state of n and the general wave function containing all the possible states. This simple expression, to our view, could be playing a central role in analyzing consciousness/mind operations because as it is clear from the equation any specific decision or behavior of the consciousness/mind; ψn(x,t); is very strongly related to the general state of the consciousness/mind; ψ(x,t); determined primarily by the instant information input via internal and external signals, previous experience/memory and environmental effects.

6. General Remarks

Physical properties of any quantum mechanical particle with a mass of m, total energy of E and potential energy of V(x) is determined by the well defined SWE that is (15). The solution of the equation gives us possible ψn(x) and En values. Time



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dependence of the wave function, for constant total energy, is standard by

0

0

−=

Ei t

(t ) .eψ ψ .

Then finding the quantum mechanical particle at a time of t and point of x for the quantum state of n is given by Pn(x,t)= ψn

*(x,t).ψn(x,t). The state or energy probability of the particle, on the other hand,

is given by Pn=2

nc . In order to give an

example, consider we have 100 identical quantum particles with a mass of m, potential energy of V(x) and total energy of E0. We experimentally carry out 100 independent measurements and measure the total energies with an experimental set up 100 times, and we get the following values; 8 times E1, 20 times E3, 10 times E5, 40 times E8, and finally 22 times E9 then the probability values for each quantum state or energy level can be calculated by using the equation (24) as;

2 22 2

1 3 5 8

2

9

8 20 10

100 100 10040 22

100 100

= = =

= =

c , c , c , c

, c

The sum of the probabilities must be equal to unity as must be in any statistical situation and it is clearly seen that the numbers above satisfying the equation (24), hence

21 +c 2

2 +c 23 +c ……..+ 2 +nc …..=1. If we

want to calculate finding probability of the particle at a quantum state of say n=3 and specific point of say x=L then we get the product of the two, in other words, the total probability is calculated by

3

20

100( L)ψ .

It seems very clear that before doing any measurement, the particle can be at any state and position with certain probabilities. However at the measurement instant, the

quantum particle is observed at a definite state n and point x and we observe this as the “physical reality”. Its is suggested that the observed state is being reality due to “environment induced selection” as suggested by (Zurek, 1981; 1982; Paz and Zurek, 1999) through the “decoherence principle” and by the process of “wave function collapse/reduction” (Tegmark, 2000). This situation can be thought analogues to the “tossing a coin” experiment. Before the experiment, the probabilities of having head or tail are equal, that is

20 5= =head headP c , and similarly

20 5= =tail tailP c , . The overall wave function

is then given by (19) as

0 5 0 5= +head tail( x,t ) , ,ψ ψ ψ .

However, at the instant of measurement, there are no probabilities, and instead there are certainties as either head or tail, then after the measurement either

21= =head headP c and

20= =tail tailP c and the

observed state wave function is ψhead or 2

0= =head headP c and 2

1= =tail tailP c and the

observed wave function is ψtail. We classically observe this final situation but no probabilities.

Consciousness/mind seems to be fitting very well to this very scientific approach in the sense that we always have choices in front of us to be selected. When we have a decision, we obviously cause the collapse/reduction of the wave function accompanying the actual quantum mechanical particle that is “Ton”. This process is then very likely to fire a specific neuron at a certain time and position, initiating whole sequence of processes of any brain or consciousness/mind activity. The basics of the subject attend but there seems to be a long way to go, however, what we say in Turkish “Starting means half way to finishing”.



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