User:Patrick0Moran/Hanyu Heisenberg

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完整的量子力學


, a physicist who was still a new professor on the tenure track at the time, succeeded in giving a quantum theoretical answer to the question of what the intensities of the lines of the hydrogen spectrum are. 當他初當助教授時，Werner Heisenberg 成功地解釋了氫燈（類似氖燈）所發光譜各條的光量. He benefited from his close association with Neils Bohr and members of his school. 因為他和Neils Bohr及其研究所的好幾位科學家本來有很友好的關係，他在這方面得了很多的幫助. From Bohr he learned the value of avoiding facile theoretical formulations that may give correct answers in some narrow context but provide no means by which to connect them to a wider context. 他從Bohr學到一個很重要的道理：要避免接受任何容易盲從的但不切實際的論點. 物理學理論不但應該在窄範圍有正確性，並且應該與廣範圍的物理學理論有其一致性. Heisenberg recalled, "Bohr would always say, 'First we have to understand how physics works; only when we have completely understood what it is about can we hope to represent it by mathematical schemes.' " (Heisenberg, Conversations, p. 230) Heisenberg回憶到，“ Bohr常常跟我們說：、我們得先了解到一個在實際上的物理過成，然後，等到我們充分地處理過這些知識才有希望用數學方式把物理表示清楚. 、" Second, from his work with one of Bohr's senior students, Hendrik Anthony Kramers, he learned the value of using difference equations in describing quantum phenomena. 次之，他跟Bohr的一個高足， Hendrik Anthony Kramers，學到在解釋電子力學現象時用“相差等式“的好處. (One difference equation has already appeared in this discussion, in the formula shown above that predicts wavelengths by multiplying a constant by the difference between two fractions.) （一個相差等式已經出現於上面的圖案中，乃 $$\frac{1}{\lambda} = R_\mathrm{H}\left(\frac{1}{m^2} - \frac{1}{n^2}\right)$$ . To make a long and rather complicated story short, Heisenberg used the idea that since classical physics is correct when it applies to phenomena in the world of things larger than atoms and molecules, it must stand as a special case of a more inclusive quantum theoretical model. 畧而言之，Heisenberg利用到"量子力學的結論不得在經典物理學範圍中不合乎經典物理學的結論"這個道理. 反過來講，比較廣泛的量子力學必能包括比較窄範圍的經典物理學的結論. 唯獨在其特殊的範圍中，量子力學才會有不合乎經典力學的結論.

So he hoped that he could modify quantum physics in such a way that when the parameters were on the scale of everyday objects it would look just like classical physics, but when the parameters were pulled down to the atomic scale the discontinuities seen in things like the widely spaced frequencies of the visible hydrogen bright line spectrum would come back into sight. 憑此道理，他有希望修改經典物理學的理論，讓新的理論在講日常生活裡的現象時，結論就與經典力學的結論完全一致，但在講原子境界時，那麼那些不連續現象，舉如氫氣光的隔離的幾個薄率等等，就會再次出現.

By means of an intense series of mathematical analogies that some physicists have termed "magical," Heisenberg wrote out an equation that is the quantum mechanical analog for the classical computation of intensities. 用了連起來的幾個似乎是魔術的類推Heisenberg創造了一個可以代替經典物理學上的光量公式的量子力學類比.

Remember that the one thing that people at that time most wanted to understand about hydrogen radiation was how to predict or account for the intensities of the lines in its spectrum. 要記住，當時研究氫燈光譜的物理學家最需要加以解釋的是其中每一頻率的光量.

Although Heisenberg did not know it at the time, the general format he worked out to express his new way of working with quantum theoretical calculations can serve as a recipe for two matrices and how to multiply them. 當時，年輕的Heisenberg並不知道所謂『矩陣』是甚麼. 可是，他創出來的新量子力學的等式在原則上可以當作兩個矩陣的藍圖，也可以指示學者怎麼算出其乘積.


 * $$C(n,n-b) = \sum_{a}^{} \, A(n,n-a)B(n-a,n-b)$$

This general format indicates that some term C is to be computed by summing up all of the products of some group of terms A by some related group of terms B. 這種方式比較廣泛、抽象. 它的意思是說有某種變數C等於加起來的變數ａ乘其對方變數b.

There will potentially be an infinite series of A terms and their matching B terms. 原則上有無數多的A，無數多的B，再說，每個A有其對方B. Each of these multiplications has as its factors two measurements that pertain to sequential downward transitions between energy states of an electron. 每次算出a乘b的乘積，A跟其對方B都有關於一個電子改變位置，比方說，A有關於電子從位置n移到位置n－a，B有關於電子從位置n－a移到位置n－b.

This type of rule differentiates matrix mechanics from the kind of physics familiar in everyday life because the important values are where (in what energy state or "orbital") the electron begins and in what energy state it ends, not what the electron is doing while in one or another state.

The formula looks rather intimidating, but if A and B both refer to lists of frequencies, for instance, all it says to do is perform the following multiplications and then sum them up:

Multiply the frequency for a change of energy from state n to state n-a by the frequency for a change of energy from state n-a to state n-b. and to that add the product found by multiplying the frequency for a change of energy from state n-a to state n-b by the frequency for a change of energy from state n-b to state n-c, and so forth: Symbolically that is: f(n, n-a) * f(n-a,n-b)) + f(n-a,n-b) * f(n-b,n-c) + etc.

It would be very easy to do each individual step of this process for some measured quantity. For instance, the boxed formula at the head of this section gives each needed wavelength in sequence. The values calculated could very easily be filled into a grid as described below. However, since the series is infinite, nobody could do the entire set of calculations.

Heisenberg originally devised this equation to enable himself to multiply two measure of the same kind, so it happened not to matter which order they were multiplied in. Heisenberg noticed, however that if he tried to use the same schema to multiply two variables, such as momentum, p, and displacement, q, then "a significant difficulty arises." It turns out that multiplying a matrix of p by a matrix of q gives a different result from multiplying a matrix of q by a matrix of p. It only made a tiny bit of difference, but that difference could never be reduced below a certain limit, and that limit involved Planck's constant, h. More on that later. Below is a very short sample of what the calculations would be, placed into grids that are called matrices. Heisenberg's teacher saw almost immediately that his work should be expressed in a matrix format because mathematicians already were familiar with how to do computations involving matrices in an efficient way.


 * $$Y(n,n-b) = \sum_{a}^{} \, p(n,n-a)q(n-a,n-a)$$ (Equation for the conjugate variables momentum and position)

Matrix of p

Matrix of q

The matrix for the product of the above two matrices as specified by the relevant equation in Heisenberg's 1925 paper is:

Where: A=p(n→n-a)*q(n-a→n-b)+p(n→n-b)*q(n-b→n-b)+p(n→n-c)*q(n-c→n-b)+.....

B=p(n-a→n-a)*q(n-a→n-c)+p(n-a→n-b)*q(n-b→n-c)+p(n-a→n-c)*q(n-c→n-c)+.....

C=p(n-b→n-a)*q(n-a→n-d)+p(n-b→n-b)*q(n-b→n-d)+p(n-b→n-c)*q(n-d→n-d)+.....

and so forth.

If the matrices were reversed, the following values would result:

A=q(n→n-a)*p(n-a→n-b)+q(n→n-b)*p(n-b→n-b)+q(n→n-c)*p(n-c→n-b)+..... B=q(n-a→n-a)*p(n-a→n-c)+q(n-a→n-b)*p(n-b→n-c)+q(n-a→n-c)*p(n-c→n-c)+..... C=q(n-b→n-a)*p(n-a→n-d)+q(n-b→n-b)*p(n-b→n-d)+q(n-b→n-c)*p(n-d→n-d)+.....

and so forth.

Note how changing the order of multiplication changes the numbers, step by step, that are actually multiplied.

Heisenberg's groundbreaking paper of 1925 neither uses nor even mentions matrices. Heisenberg's great advance was the "scheme which was capable in principle of determining uniquely the relevant physical qualities (transition frequencies and amplitudes)" of hydrogen radiation.

Paul Dirac decided that the essence of Heisenberg's work lay in the very feature that Heisenberg had originally found problematical -- the fact of non-commutativity such as that between multiplication of a momentum matrix by a displacement matrix and multiplication of a displacement matrix by a momentum matrix. That insight led Dirac in new and productive directions.