(1-1)

(1-2)

(1-3)

Quantum theory attempts to solve Eq.(1-1) by successive approximation. The series expansion of is

(1-4)

(1-5)

(1-6)

Firstly, we find out the expression of for arbitrary N. Secondly, we find up the sum of all to obtain , and test that is the exact solution of Eq.(1-1). Finally we calculate the transition probability from . This program of discussion is close to the way used by quantum mechanics. Naturally, the conclusion of unitary matrix equation is same as the one of the Schrödinger equation.

Now we prove that the expression of is

(1-7)

by the mathematical inductive method. In which and are defined as

(1-8)

(1-9)

(1-10)

and satisfy the set of equation

(1-11)

(1-12)

(1-13)

(1-14)

1. If

(1-15)

    (1-16)

This expression can be rewritten as

(1-17)

where we use Eqs. (1-11), (1-12) and

Therefore the expression (1-7) is the solution of Eqs. (1-5) and (1-6) for , if it is the solution for . We have proved that this conclusion is correct for , so it is correct for arbitrary .

　

1-2 Expression of

Exchanging the order of summation, Eq. (1-7) can be rewritten as

(1-18)

In order to obtain the expression of , we add up the infinite series

(1-19)

which

(1-20)

(1-21)

From Eqs. (1-8)---(1-11), , are the eigenvalue and eigenfunction of the equation of stationary-state respectively,

(1-22)

and the eigenfunctions satisfy the orthonormality conditions

(1-23)

(1-24)

Substituting Eq. (1-19) into Eq. (1-1) and using Eq. (1-22), we can prove that it is the exact solution of Eq. (1-1)

(1-25)

The important point is that this conclusion is correct in arbitrary region of .

1-3 Transition probability per unit time

　

Now let us compare the transition probability per unit time deduced from the exact solution and the one from the sum of first finite orders approach .

For clear, some book [6] define and separately. The transition probability per unit time is

(1-26)

where

(1-27)

and . According to the exact solution, i.e. Eq.(1-19), we have

(1-28)

Eq. (1-23) means that is the -component of one unit vector, and is the -component of another unit vector. The expression

is the product of two unit vectors, so

(1-29)

Finally we obtain

(1-30)

This is the result deduced from the exact solution of Eq. (1-1). Obviously, it cannot be used to describe the transition processes.

Now let us consider the transition probability per unit time deduced from the successive approximation. As an example, we only take the first order approximation

(1-31)

into account, which

(1-32)

(1-33)

(1-34)

(1-35)

then we obtain

(1-36)

  (1-37)

(1-38)

It is the transition probability per unit time deduced from . From Eq. (1-19), we can understand that the substance of successive approximation is only to expand and in the expression of according to the stationary perturbation Eqs. (1-8)---(1-14). There is an unavoidable basic problem for any approach, how much is the difference between the approach solution and the exact solution? If this difference is large than the experiment error, then the comparison between the approximate solution with the experiment will be nonsensical. If we demand the approach method is significant, the difference between the sum of first finite order of approximate solutions with the exact solution must be less than a positive number , it may be expressed in an accurate mathematical form

(1-39)

For a given value (>0), we can find up a number , that this relation satisfies. It means that the series is convergence. In which region is the series convergence? However, in the region , the expansion series of stationary perturbation is divergent. satisfies Eq. (1-23), but satisfies Eq. (1-35), it is wrong in the region . Quantum mechanics just uses the factor to obtain the function in Eqs. (1-37) and (1-38). The error of successive approximation is using the sum over the first finite terms to substitute in the divergent region of series.

The equation of unitary matrix cannot be used to describe the transition processes, as the Schrödinger equation.

　

2 Discussion on the method of Green’s function

Dyson pointed out that satisfies the integral equation

(2-1)

where Green’s function is defined as

(2-2)

and , are the eigenvalue and eigenfunction of , respectively.

In quantum mechanics, people use the method of successive approximation to solve Eq.(2-1). Suppose that

(2-3)

where

(2-4)

(2-5) 

This discussion is in the continuous case, because of Eq. (2-2) is integral over the continuous variable .

Our task is to find and from Eqs. (2-4), (2-5) and (2-3). The procedure of inference is the same as we do in section 1.

　

2-1 Expression of

　

Now we prove that the expression of is

(2-6)

where , are the defined by Eqs. (1-8)—(1-14).

1. If , Eq.(2-6) becomes

(2-7)

Therefore, if , Eq.(2-6) is just Eq.(2-4).

2. Suppose that Eq.(2-6) is correct for , then we have

   　

(2-8)

　

We first perform the -integration by contour integration. Considering the complex variable , if ,

therefore we need to close the contour in the lower half plane. In this case the pole will be included. The theorem of residua gives

　Fig.1 The contour of -integration if

(2-9)

If , we have

therefore we need to close the contour in the upper half-plane, there is no pole encompassed, so that

(2-10)

So the t’-integration becomes

(2-11)

Next, we perform the -integration and -integration

(2-12)

where we use the same manner as to prove Eq.(2-10). Finally, we perform the t’-integration and obtain

(2-13)

This expression means that Eq. (2-6) is the solution of Eq.(2-5) for , if it is the solution for . We have prove that this conclusion is correct for , so it is correct for arbitrary .

　

2-2 Expression of

In the same manner as we did in section 1-2, we obtain

　

(2-14)

It is the exact solution of Eq. (2-1). This conclusion may be proved by direct substitution

　

(2-15)

　

2-3 Transition Probability per Unit Time

　

Quantum mechanics uses to express . The relation is

(2-16)

It uses to express . The relation is

(2-17)

According Eqs. (2-13) and (2-16), we get

(2-18)

According Eqs. (2-14) and (2-17), we have

(2-19)

Therefore the conclusion of the method of Green’ function is same as the one of Schrödinger equation.

　

Summary:

1. The Schrödinger’s equation, the equation of unitary matrix and the method of Green’s function are equivalence;
2. The Fermi’s golden rule is not a mathematically reasonable deductive inference from the relative equation;
3. The exact solution of these equations cannot give the transition probability per unit time which is consentaneous with the experiments.

　

Does the hypothesis for electron can provide a picture to describe the time-dependent processes? It is the physical substance of the problem.

　

　

　

References

　

1. Zhang Junhao, Physics Essays 10,1 (1997).
2. P.A.M.Dirac, The principles of Quantum Mechanics. 3^rd. ed. (Clarendon Press, Oxford, 1947).
3. Feynman R.P., Phys.Rev. 76,749 (1949).
4. T.Y. Wu and T. Ohmura, The Quantum Theory of Scattering. (Prentic Hall, Englewood Cliffs, N.J. 1962).
5. P.roman, Advanced Quantum Theory. (Addison-Wesley, Reading, MA, 1965).
6. L.I.Schiff, Quantum Mechanics. 2ec ed. (1955).