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Longmarch
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Time as a Derived Physical Quantity
Consider a physical system whose state changes and let us call it a Dynamical System.
Let a closed Dynamical System, S, whose state be described by ψ-- A mathematical entity we use to describe S (or the state of S). ψ can take on the following specific states: ψ1,.. ψ2,.. ψi,..ψj,..ψk,... where i, j, k,.. are positive integers. Then one can define Time increment (Time or duration), δt, as follows:
Suppose S initially takes up the state ψi, and then it changes to ψj (written as ψi → ψj), then a certain time increment, δt, is said to have elapsed; where δt = 0 if i = j, and δt ≠ 0 if i ≠ j.
A Time increment defined in this way is therefore a Derived Physical Quantity, and
1) If S does not exist, then Time, δt, does not exist.
2) Time is quantized if the states ψi, ψj,... etc are discrete.
3) If ψ changes in the sequence "ψi → ψj → ψk", then the 2 "δt's" (associated with the 2 "→") are perceived as being equal in magnitude since, in reality, only ψi, ψj and ψk, but not something in-between, that are observable. It is therefore meaningless to say that the "in-betweens" can be of different magnitudes. But we are quite sure that the "in-betweens", the "δt's", do exist because their end-points exist.
4) If ψ changes in sequence given by "ψi → ψk → ψj", where i, j and k all have different values, then the 2 "δt's" are taken as having the same sign (direction). This follows naturally from the way "δt" is defined above since the 2 "δt's" are defined in exactly the same manner, if one has a positive (or negative) sign, so does the other.
Following the above, if ψ changes in sequence given by "ψi → ψk → ψj → ψi", we should perceive S having returned to its previous state of ψi, instead of perceiving as Time having gone backwards. Time, therefore, is unidirectional.
Recall the Definition: A Dynamical System is a Physical System whose state changes. Nevertheless, we still want to ask the question: Why does it change? Why is it not non-changing and stay the same always?
The fact that it changes allows us to postulate the existence for some physical agent Α, which when acts on S at ψi, makes possible of the appearance of S at ψj, and accompanying such a change, there is an elapse of time δt-- By definition.
Hence, Α : Sψi → Sψj where i ≠ j--------------- [S1]
Now, S denotes the Dynamical System, ψ is a mathematical entity we use to describe S (or the state of S), i, j, k,... etc are labels we attach to ψ in order to distinguish the different states of S.
From statement [S1], it is reasonable to assume the existence of a corresponding mathematical operator Â, of A, such that
Âψi = ψj--------------------------------- [E1]
This is merely rewriting statement [S1] in equation form by introducing the operator Â.
After acting on S at ψi, A persists-- There is no reason why it should vanish suddenly, especially in a closed system-- and continues to act on S at ψj. Hence we have
Α : Sψj → Sψk where j ≠ k--------------- [S2]
Α : Sψk →Sψl where k ≠ l--------------- [S3]
Α : Sψl →Sψm where l ≠ m------------- [S4]
and so on.
From these we can get
Âψj = ψk--------------------------------- [E2]
Âψk = ψl--------------------------------- [E3]
Âψl = ψm ------------------------------- [E4]
and so on.
Given the set of equations [E1], [E2], [E3],..., can we say anything about Âψ?
Is it correct to say that Âψ = aψ, where "a" is a function independent of ψ?
Let ℤ be the set {ψ1, ψ2,... ψi,...ψj,...ψk,...}
ℤ = (All possible ψ's}
So, Â maps each ψ in ℤ into another ψ in ℤ and, by inspection, it looks plausible that the Âψ = aψ is a solution to the set of equations [E1], [E2], ...
The author cannot give a rigorous mathematical proof that this is so but maybe a more concrete example could show that this is highly plausible:
Let ℂ be a set of all boys in a class
ℂ = {All b's in a class), where b = boy
Specifically, the boys are Stephen, John, Martin, Robert, Ian, Phillip, David, Charles, Allan,...
Therefore,
ℂ = {s, j, m, r, i, p, d, c, a,... }
Now, T is the teacher who is directing the boys to play a game of passing on a relay-stick. When one particular boy gets the stick, the teacher decides to whom it should be handed to according to a set of rules the teacher has in his head.
We do not yet know what that set of rules is but we can note down how the relay-stick is passed along among the boys. For example, at some stage Martin is holding the stick and the teacher instructs him to pass it to Phillip. So,
Tm = p--------------- (1)
And then the teacher instructs Phillip to pass it to Robert and so on...
Tp = r--------------- (2)
Tr = s--------------- (3)
Ts = a --------------- (4)
and so on.
From the above, what can we say about "Tb"? We know that it has to be a "b" (boy) but there are constraints. The only way to make an consistent equation is to have
Tb = (attachment)b
In this particular example, the attachment might be a description of the set of rules the teacher uses to decide how the boys should pass the stick along. If the rules are known, all the equations (1), (2), (3), (4),... can be worked out without actually following the passing of the relay-stick. Therefore, the equation Tb = (attachment)b is equivalent to the whole set of equations (1), (2), (3), (4),...
Now go back to Âψ. By analogy, it seems reasonable to assume that the operator equation, Âψ = aψ, where the attachment is "a", is the solution to (the equivalent of) the set of equations {E1], [E2], [E3],... What's more, from what we already know about operator equations, the relationship between the attachment "a" and ψ should be one of multiplication.
Assuming that the above arguments are valid it appears that, just from the definition of a Dynamical System being a physical system that changes, one can show that the agent responsible for that change, the physical agent A, should have a corresponding operator  (or such an operator can be constructed) that is related to ψ, the mathematical entity used to describe the physical system, in the form of an operator equation, Âψ = aψ, where "a" should reflect the nature of Â, and therefore of A. In other words, the equation Âψ = aψ arises for the fact that the physical system changes!
Knowing that total energy normally determines how a physical system evolves, it is therefore reasonable to postulate that A should be the Hamiltonian and  the Hamiltonian operator.
Consider a physical system whose state changes and let us call it a Dynamical System.
Let a closed Dynamical System, S, whose state be described by ψ-- A mathematical entity we use to describe S (or the state of S). ψ can take on the following specific states: ψ1,.. ψ2,.. ψi,..ψj,..ψk,... where i, j, k,.. are positive integers. Then one can define Time increment (Time or duration), δt, as follows:
Suppose S initially takes up the state ψi, and then it changes to ψj (written as ψi → ψj), then a certain time increment, δt, is said to have elapsed; where δt = 0 if i = j, and δt ≠ 0 if i ≠ j.
A Time increment defined in this way is therefore a Derived Physical Quantity, and
1) If S does not exist, then Time, δt, does not exist.
2) Time is quantized if the states ψi, ψj,... etc are discrete.
3) If ψ changes in the sequence "ψi → ψj → ψk", then the 2 "δt's" (associated with the 2 "→") are perceived as being equal in magnitude since, in reality, only ψi, ψj and ψk, but not something in-between, that are observable. It is therefore meaningless to say that the "in-betweens" can be of different magnitudes. But we are quite sure that the "in-betweens", the "δt's", do exist because their end-points exist.
4) If ψ changes in sequence given by "ψi → ψk → ψj", where i, j and k all have different values, then the 2 "δt's" are taken as having the same sign (direction). This follows naturally from the way "δt" is defined above since the 2 "δt's" are defined in exactly the same manner, if one has a positive (or negative) sign, so does the other.
Following the above, if ψ changes in sequence given by "ψi → ψk → ψj → ψi", we should perceive S having returned to its previous state of ψi, instead of perceiving as Time having gone backwards. Time, therefore, is unidirectional.
Recall the Definition: A Dynamical System is a Physical System whose state changes. Nevertheless, we still want to ask the question: Why does it change? Why is it not non-changing and stay the same always?
The fact that it changes allows us to postulate the existence for some physical agent Α, which when acts on S at ψi, makes possible of the appearance of S at ψj, and accompanying such a change, there is an elapse of time δt-- By definition.
Hence, Α : Sψi → Sψj where i ≠ j--------------- [S1]
Now, S denotes the Dynamical System, ψ is a mathematical entity we use to describe S (or the state of S), i, j, k,... etc are labels we attach to ψ in order to distinguish the different states of S.
From statement [S1], it is reasonable to assume the existence of a corresponding mathematical operator Â, of A, such that
Âψi = ψj--------------------------------- [E1]
This is merely rewriting statement [S1] in equation form by introducing the operator Â.
After acting on S at ψi, A persists-- There is no reason why it should vanish suddenly, especially in a closed system-- and continues to act on S at ψj. Hence we have
Α : Sψj → Sψk where j ≠ k--------------- [S2]
Α : Sψk →Sψl where k ≠ l--------------- [S3]
Α : Sψl →Sψm where l ≠ m------------- [S4]
and so on.
From these we can get
Âψj = ψk--------------------------------- [E2]
Âψk = ψl--------------------------------- [E3]
Âψl = ψm ------------------------------- [E4]
and so on.
Given the set of equations [E1], [E2], [E3],..., can we say anything about Âψ?
Is it correct to say that Âψ = aψ, where "a" is a function independent of ψ?
Let ℤ be the set {ψ1, ψ2,... ψi,...ψj,...ψk,...}
ℤ = (All possible ψ's}
So, Â maps each ψ in ℤ into another ψ in ℤ and, by inspection, it looks plausible that the Âψ = aψ is a solution to the set of equations [E1], [E2], ...
The author cannot give a rigorous mathematical proof that this is so but maybe a more concrete example could show that this is highly plausible:
Let ℂ be a set of all boys in a class
ℂ = {All b's in a class), where b = boy
Specifically, the boys are Stephen, John, Martin, Robert, Ian, Phillip, David, Charles, Allan,...
Therefore,
ℂ = {s, j, m, r, i, p, d, c, a,... }
Now, T is the teacher who is directing the boys to play a game of passing on a relay-stick. When one particular boy gets the stick, the teacher decides to whom it should be handed to according to a set of rules the teacher has in his head.
We do not yet know what that set of rules is but we can note down how the relay-stick is passed along among the boys. For example, at some stage Martin is holding the stick and the teacher instructs him to pass it to Phillip. So,
Tm = p--------------- (1)
And then the teacher instructs Phillip to pass it to Robert and so on...
Tp = r--------------- (2)
Tr = s--------------- (3)
Ts = a --------------- (4)
and so on.
From the above, what can we say about "Tb"? We know that it has to be a "b" (boy) but there are constraints. The only way to make an consistent equation is to have
Tb = (attachment)b
In this particular example, the attachment might be a description of the set of rules the teacher uses to decide how the boys should pass the stick along. If the rules are known, all the equations (1), (2), (3), (4),... can be worked out without actually following the passing of the relay-stick. Therefore, the equation Tb = (attachment)b is equivalent to the whole set of equations (1), (2), (3), (4),...
Now go back to Âψ. By analogy, it seems reasonable to assume that the operator equation, Âψ = aψ, where the attachment is "a", is the solution to (the equivalent of) the set of equations {E1], [E2], [E3],... What's more, from what we already know about operator equations, the relationship between the attachment "a" and ψ should be one of multiplication.
Assuming that the above arguments are valid it appears that, just from the definition of a Dynamical System being a physical system that changes, one can show that the agent responsible for that change, the physical agent A, should have a corresponding operator  (or such an operator can be constructed) that is related to ψ, the mathematical entity used to describe the physical system, in the form of an operator equation, Âψ = aψ, where "a" should reflect the nature of Â, and therefore of A. In other words, the equation Âψ = aψ arises for the fact that the physical system changes!
Knowing that total energy normally determines how a physical system evolves, it is therefore reasonable to postulate that A should be the Hamiltonian and  the Hamiltonian operator.