Quantum mechanics take the sum of first finite order approximate solutions of time-dependent perturbation to substitute the exact solution, it will produce the error. How much is this error? However people do not find up for arbitrary N and the exact solution , so quantum mechanics can not answer this question. Therefore this ‘unknown error’ is as a premise of all theoretical conclusions of quantum mechanics for the transition processes. This problem is relative to the question. Can we really use the Schrödinger equation to describe the transition processes? So it is the most important unsettling problem of physical theory in 20^th century. All of physical theory cannot avoid the strictly mathematical inference.

We find out the time-dependent perturbation approximate solution for arbitrary order and the exact solution . Then we can prove that: (1) In the neighborhood of energy conservation, the series is divergent. The basic error of quantum mechanics is using the sum of the first finite order approximate solutions to substitute the exact solution in this divergent region. It leads to an infinite error. So the Fermi’s golden rule is not a mathematically reasonable inference of the Schrödinger equation (2) The transition probability per unit time deduced from the exact solution of Schrödinger equation cannot describe the transition processes.

(1)

(2)

We may write the as the product of time independent with a time dependent factor . Quantum mechanics only discusses two types , they are

(3)

(4)

The eigenvalues of are discrete or continuous. We shall discuss for each case. In discrete case, according to quantum mechanics, suppose

(5)

(6)

(7)

(8)

(9)

So far, people only find the few order approximate solutions of (7). We do not find out for arbitrary order. The only way is to substitute the exact solution by the sum of first finite order approximate solutions

(10)

The problem is that, whether the may be abandoned? From the point of physics, if the abandoned part is large then the experimental error, then we cannot obtain any conclusion from the comparison between the sum of first finite order approximate solutions with experimental results. In this case, the words “Schrödinger equation may describe the transition processes” would be meaningless. From the point of mathematics, is the function of energy . Only in the convergent range of series, for , we can find a number M , that the relation

(11)

is satisfied. What is for arbitrary order? What is the exact solution ? In which range is the series divergent? These questions are relative to that, can the equation of quantum mechanics describe the transition processes? So it is the most important unsettled question of physical theory.

The equations (6) and (9) contain , it means that we discuss in the discrete case. We shall solve the Schrödinger equation and its approximate equations in the range .

(12)

(13)

(14)

(15)

to substitute in the expression of , then it is easy to discover the general form of the time dependent perturbation approximate solution. and satisfies the stationary state approximate equations

(16)

(17)

where satisfies the orthonormalization condition

(18)

In appendix A we prove that the order time dependent perturbation approximate solutions may be expressed as

(19)

(20)

The first order of time dependent perturbation approximate solution deduced by quantum mechanics does not contain the term. This term is

(22)

It is just the term in Eq. (20). So you see the result give by quantum mechanics is incomplete. The incompleteness in the low order approximate solution will influence the calculation in the high order approximate solution.

(23)

(24)

(25)

We shall take in the latter calculation, so we have

(27)

According to Eqs.(24)—(26), you may guess that the general form of must be

(28)

Because the first order time dependent perturbation approximate solution deduced by quantum mechanics is incomplete, then the high order approximate solution will do not contain the term relative to . For comparing, let us consider the first term of Eq.(23),

(29)

This formula contains all of orders approximate solution deduced by quantum mechanics. For example, in the case of , we obtain

(30)

(31)

The difference between these two equations is due to that quantum mechanics neglects the contribution of in calculation the first order approximate solution. The simple formula of Eq.(29) not only contain all order approximate solutions that they have been found out. It is also correct in the case for arbitrary order.

Other coefficients are

(32)

(33)

(34)

(35)

where is the function of , it is not relative to .The recurrence relation is

(36)

(37)

is the sum of many terms, each term contains factors of

(38)

The total of approximate order is

(39)

Summarizing above results, the expression of for arbitrary order is

We prove this formula in the case , but it is only a guess in the high order case. However the task focus on the proposition, if Eq.(40) is correct for order, try to prove that it is correct for order. We prove this proposition in appendix B.

(42)

You can find the calculating process in appendix C. The opposite process is to expand presented in Eq.(42) by using the stationary-state perturbation approximation, we may obtain the different order of time-dependent perturbation approximate solution .

Finally we test that is the exact solution of the Schrödinger equation in appendix D. It means that this conclusion is correct not only in the convergent intervals, but also correct in the divergent intervals of the series.

(ii) Expanding the coefficient presented in according to the stationary

state perturbation, and classifying according to the different approximate order we get

;

(iii) is satisfied the time-dependent perturbation approximate equation and special initial condition.

The exact solution must be expressed by the solution of stationary state equation . Substituting the stationary state perturbation approximate solution into presented in the exact solution, we shall obtain the different order approximations of the time–dependent perturbation method . Therefore, the divergent intervals of the time-dependent perturbation approach and the one of the stationary state perturbation approach are identical. Quantum mechanics pointed out that the divergent interval of the stationary state perturbation series is . So it must be the divergent interval of the time-dependent perturbation series. It means that, for each , no matter how much order approach solutions we take into account, the relation

does not satisfy in the divergent interval. We cannot use the sum of first finite order of time-dependent perturbation solutions to substitute the exact solution. The error of time-dependent perturbation method is using to substitute for calculating the transition probability in the divergent range . Let us examine the first-order approximation of the time-dependent perturbation

is the -component of an unit vector. Therefore

(44)

and Eq.(43) can only be used in the region .

(45)

(46)

There are two questions. First, the premise of this section is that we discuss in the discrete case. Quantum mechanics changes to discuss the continuous case and simply substitutes Eq.(45) with Eq.(46). Is this reasonable? We shall discuss this question in the next section. Second, the transition probability deduced from the perturbation approximation is correct only in the region ; the fault is in . If we use the expression of the first order stationary state perturbed approach in the region , the error is

If .

Quantum mechanics discuss the relation between with . And pointed out there is a peak of function in , this peak of function is just deduced from the error factor . If , the high of peak tends to infinite,and the breadth of the peak tends to zero, the discussed function becomes function. However if

, the whole main peak of function is in the divergent region of the perturbation approximate series. The problem is that it is under these two conditions, , and

Fig.1 The relation between with , where I is the

, that quantum mechanics obtains

(47)

from Eq.(46). Fermi called it as the golden rule. Of course, if this formula is correct, then someone can find a suitable to be identical with the experimental value , but mathematically Fermi’s golden rule is not a reasonable conclusion from the Schrödinger equation.

(48)

(49)

It means that is the -component of a unit vector, and is the -component of another unit vector. The expression

(50)

(51)

This is the result deduced from the exact solution of Schrödinger equation (1) in the case. This result cannot be used to describe the transition processes.

1. The substance of the time-dependent perturbation approximation is to expand presenting the expression of according to the stationary-state perturbation approach..
2. The convergent interval of the stationary-state perturbation approach is , this region is also the convergent interval of the time-dependent perturbation approach..
3. The sum of the first finite order perturbation approximate solutions may be used to substitute the exact solution only in the convergent interval. The fundamental error of the time-dependent perturbation approach is using the sum of the first finite order approximate solutions to calculate the transition probability in the divergent interval . Mathematically Fermi’s golden rule is not a reasonable conclusion from the Schrödinger equation in the discrete case.
4. People use the sum of first finite order perturbation approximate solutions to substitute the exact solution. It only caused by that they did not find out the approximate solution for arbitraryorder. When we find out and , then we have not any reason to use the sum of first finite order perturbation approximate solutions to substitute the exact solution. The pity is that, the transition probability per unit time deduced from the exact solution of the Schrödinger equation is zero in the discrete case, it cannot be used to describe the transition processes.

In the continuous case, the discrete variable must change to continuous variable , the summation must change to the integration, and presented in the discrete case must change to , the others are same as in the discrete case. This kind of variation cannot give the difference result.

(52)

(53)

If the initial condition takes the form of Eq.(53), then at the probability of the system in the region is

(54)

(55)

where are the eigenvalue and eigenfunction of the stationary-state equation

(56)

(57)

(58)

(59)

where satisfy a set of equations

(60)

(61)

(62)

(63)

The relations between with are same as in the discrete case.

are the solution of the time-dependent perturbation approximate equations

(65)

(66)

(67)

,

(68)

(69)

However, at , the probability of the system in is.

(70)

The transition probability in is a relative probability. It is defined as

(71)

(72)

(73)

In the continuous case, satisfies the orthonormalization condition (57). Then is the -component of a normalization vector, and is the -component of another normalization vector. The expression

(74)

(75)

(76)

(78)

(79)

where is the probability amplitude of the state in which there is an electron in and photons of energy . If at , the system is in the state then

(80)

(81)

which , are the solution of the stationary-state equation

(82)

(83)

(84)

At the same time, meets the orthonormalization conditions of the eigenfunctions:

(85)

(86)

Obviously, the exact solutions solved under the cases and have the identical mathematical structure. The results in Sec. 2 are still applicable in this case.

1. In the cases in which the time factors are or , the eigenvalues of are discrete or continuous, the process to deduce the transition probability per unit time from time-dependent perturbation approximation contains a basic mathematical mistake. It is using the sum of first finite order approximate solutions to substitute the exact solution for calculating the transition probability in the divergent range—the neighborhood of energy conservation. Fermi’s golden rule is not the mathematically reasonable deductive inference from the Schrödinger equation.
2. The transition probability per unit time deduced from the exact solution of the Schrödinger equation cannot be used to describe time-dependent processes.

(87)

(88)

Quantum mechanics found out the first finite order time dependent perturbation solutions, but it is difficult to look for the general form of the high order solution from them. We introduce a new method to calculate the approximate solution and obtain a new expression. It is more complex then the usual way, but you can easy to look for the general form of arbitrary order approximate solution from them.

(A1)

(A2)

(A3)

(A4)

(A5)

(A6)

(A7)

(A8)

(A9)

(A10)

(A11)

(A12)

(A13)

(A14)

We add a term to the second line of this expression. According to Eq. (A1), the contribution of this term is zero. We shall explain it in latter. The definition of is

(A15)

(A16)

Suppose that has the form

(A17)

(A18)

(A19)

(A20)

(A21)

(A22)

The first order time dependent perturbation approximate solution deduced from quantum mechanics does not contain the term , this term is

(A23)

It is corresponding to the term in Eq. (A17).

(A24)

therefore satisfies the special initial condition. It is the reason that we add a term in Eq.(A14).

(A25)

(A26)

(A27)

in Eq. (A26). The definition of is

(A28)

(A29)

(A30)

The right side of this expression is sum for all . When , it has the form . So the expression of contains the term of . So we suppose that the solution has the form

(A31)

(A33)

(A34)

(A35)

(A36)

(A37)

(A38)

(A39)

satisfies the initial condition.

(A10)

(A17)

(A31)

The coefficients decompose two sets, one set is , and another set is . For the set , we have

(A11)

(A22)

(A36)

(A40)

(A41)

(A21)

(A38)

(A37)

Then we may image that, in the case

(A42)

where is only the function of . The first few functions are

(A43)

(A44)

(A45)

We shall obtain more knowledge of in the latter,

(A46)

(A47)

is the sum of many terms, each term contains factors of ,

(A48)

Appendix B: The time-dependent perturbation approximate solution is Eq.(40) for arbitrary N order

(B1)

(B2)

(B3)

(B4)

(B5)

The stationary state equation is
(B6)
Its perturbation approximate equations are
(B7)
(B8)
So we get
(B9)
We obtain that
the first term of (B9) (B10)

because of
(B11)
The second term of (B9) is relative to . The region of two sums is in 

Fig.2.

Fig. 2 The region of two sums is on the triangle (containing the boundary).

We may exchange two sums by this way

and obtain that
the second term of (B9) (B12)
Then the relation
 (B5)
is correct.
Now we use the inductive method to prove the time-dependent perturbation approximate solution for arbitrary N order is

(40)
If , (40) may be simplified as
(B13)
It is the solution of zeroth order time-dependent perturbation equation and satisfies the special initial condition. If , (40) may be simplified as

(B14)
It is the solution of the first order time-dependent perturbation equation and satisfies the initial condition.
Supposing that the solution of N order perturbation equation is (40), we shall prove that the solution of N+1 order perturbation equation has the form same as (40).


(B15)

The first term of (B15) =

A factor in second term of (B15) is

Fig.3 Exchanging two sums in above expression, .

Where
(B16)
Therefore,
the second term of (B15)

Substituting above results into (B15), we get
(B17)
Now let us prove that the expression of is
(B18)
In fact,
(B19)
So (B17) satisfies the N+1 order perturbation approximate equation. In addition,
(B20)
Therefore satisfies the special initial condition. So we prove that, if (40) is correct for N order, then it must be correct for (N+1) order. We have prove this expression is correct for N=0,1orders. So it must be correct for arbitrary N order. 

Appendix C: The expression of exact solution

We have obtain

(C1)

Fig. 4 Exchanging two sums in Eq.(C1), .

According to the time-dependent perturbation approximate method, the exact solution is defined as
(C2)
Now the task is to calculate the sum in this expression. Notice that the contribution of second term of Eq.(C2) in the case is zero. Therefore
(C3)
Suppose
(C4)
then we have 
(C5)
Using this relation, (C2) may be written as
(C6)

Fig.5 Exchanging two sums in Eq.(C6), .

We prove that the relation
(C7)
is correct.
According to (B2), we get
(C8)
so (C7) is correct for . Suppose that (C7) is correct for , then we have 
(C9)
Therefore (C7) is correct for . The conclusion is that the relation (C7) is correct for arbitrary .
According to Eq.(B1),
(C10)
Substituting Eqs.(C7) and (C10) into Eq.(C6), we finally obtain
(C11)
It is just the exact solution of Schrödinger equation. From this deducing process, we conversely prove that, using the stationary state perturbation method to expand in exact solution, we may obtain the different order of time-dependent solutions.

Appendix D: Test the exact solution of Schrödinger equation

Now we test that, is
(D1)
the exact solution of Schrödinger equation? Does it satisfy the initial condition? In fact,
(D2)
Therefore Eq.(D1) satisfies the Schrödinger equation. In addition, if 
(D3)
It is the special initial condition.

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Fig.2

Fig.3

Fig.4

Fig.5

References
1. P.A.M. Dirac, The Principles of Quantum Mechanics, 3rd.ed. (Clarendon Press,Oxfold, 1947)
2. R.P. Feynman, Phys. Rev. 76,749 (1949).
3. Idem, Rev. Mod. Phys. 20, 367 (1948).
4. T.Y. Wu and T. Ohmura, The Quantum Theory of Scattering (Prentice Hall, Englewood Cliffs, NJ,1962).
5. P. Roman, Advanced Quantum Theory (Addison-Wesley, Reading, MA, 1965).
6. L.I. Shiff, Quantum Mechanics, 2nd.ed. (1955).
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