to describe the process that an electron transits from one state to another state, where is the time factor. The time dependent perturbation of quantum mechanics expands as a series

 (2)

However people only find out a few terms (not more than 10 terms) of this series. So the only way is to take the sum of first few terms to substitute the whole series, but this substitution inevitably produces the error. How much is this error? Quantum mechanics do not find out for arbitrary , do not find out , so it cannot answer this question. In the strictly speaking, this ‘unknown error’ is as a premise of all theoretical conclusions of quantum mechanics for the transition processes. If this ‘unknown error’ is large enough, it will be a terrible black hole swallowing up all achievements of quantum mechanics for transition processes. There are two fundamental questions: can the sum of first finite orders of the approach substitute the exact solution? Does the error produced by this substitution permit? This is the most important unsettled question of physical theory in 20^th century. For answer this question, we must: (1) find out the expression of for arbitrary , (2) find out the expression of , (3) find out the convergent region of the time dependent perturbed approximation series. Most theoretical physicists deem that it is an insurmountable barrier. In this paper, we will answer this unsettled question. Certainly it need a long and troublesome calculation. We only introduce the new idea and the main point of this calculation.

The first step is to find out the expression of for arbitrary . In this step, the key is to find out its general formation from the first few orders of approximate solution. However the form of the low orders approximate solution is so complex, then it is difficult to find its general rule. In quantum mechanics, the approximate solution is expressed by . If using to substitute , we shall obtain the unforeseen result, where is the solution of stationary-state perturbation equation

(3)

(4)

 (5)

(6)

In all the book of quantum mechanics, it is firstly discuss the case that the time factor is , and the eigenvalues of are discrete. If we take to substitute presented itself in the expressions of first 3 orders of time dependent perturbed approximation, then we obtain

(7)

(8)

(9)

where the subscript means that the expression is deduced from the results of quantum mechanics. The expression of second order perturbation approximation deduced from quantum mechanics is a little different from (9), this fact is caused by that quantum mechanics neglects the contribution of term in .

According to these three equations, you may guess that the general expression for arbitrary must be

(10)

This expression contains all achievements of quantum mechanics for the time dependent problem and extends to the arbitrary order. Then you take a leap over the insurmountable barrier.

Next step is to sum all of

,(11)

However this expression is not the exact solution of Schrödinger equation. The mistake is caused by that there are some contributions be neglected in quantum mechanics, such as term in . Considering the contribution of all terms, the mathematical construction of (8), (9) and (10) is incomplete. It only contains the term, does not contain the term. The complete expression of order approximation solution is

(12)

(13)

We can use the mathematical inductive method to prove that (12) is correct, and find out the recurrence relation of the coefficient . However (10) is not a complete expression, so we cannot prove that it is correct by directly using the mathematical inductive method. Using the method same as (11), we obtain

(14)

Substituting this expression into Schrödinger equation, we can prove that it is just the exact solution in arbitrary energy . Also (14) satisfies the special initial condition. According to quantum mechanics, the transition probability per unit time is

(15)

The transition probability per unit time deduced from the exact solution cannot be used to describe the transition processes, but the one deduced from can be used to describe the transition processes, why so?

According to about discuss, the exact solution is correct in arbitrary region of . The substance of the time dependent perturbed approach is to expand presented in (14) according to the stationary state perturbed approach. Therefore the convergent intervals of the time dependent perturbed approach and of the stationary state perturbed approach are identical. Quantum mechanics has pointed out that the convergent interval of the stationary state perturbed approach is

(16)

(17)

Quantum mechanics uses this expression in the divergent interval of the series and obtains the Fermi’s golden rule. In this interval, the main part of the error produced by using the first order approach solution to substitute the exact solution is caused by using to substitute in (17). The late error is

(18)

If , . So the Fermi’s golden rule deduced from (17) is mistake, it is not a mathematically reasonable inference from the Schrödinger equation.

The conclusion is that, in the case time factor is and the eigenvalues of are discrete, the exact solution of Schrödinger equation cannot describe the transition processes. In the neighborhood of energy conservation, there is a great error produced by using the sum of first finite time dependent perturbed approximate solution to substitute the exact solution, so Fermi’s golden rule is not the mathematically reasonable inference from Schrödinger equation. This mistake of time dependent perturbed approach covers the face that Schrödinger equation cannot describe the transition processes.

In the case that the eigenvalues of are continuous and in the case that time factor is , the mathematical construction of Schrödinger equation is analogous to the one in above discussion. So we shall obtain the same conclusion in these two cases.

Quantum mechanics deems that we must use the wave picture to describe the motion of electron. (14) is just a standard picture that the state varies with time. The system is oscillation between all possible states, therefore the average transition probability per unit time is zero. So this picture cannot describe the transition processes. (16) is a special wave picture, it may describe the transition process by using the method of quantum mechanics. The character of this special oscillation is that: if , the amplitude tends to infinite, but it contradicts with the fundament arrange of quantum mechanics. The square of amplitude is the probability of the system occurring at a special state, but the probability is not more than 100%, it cannot tend to infinite. So this amplitude must be a mathematical mistake. This is the physical substance of above discussion.