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Konjunktiv I is mostly used for indirect speech and wishes. The following sentence is neither:

Die Funktionen ψn(x) seien Eigenfunktionen eines Hamiltonoperators Ĥ.

Why is the Konjunktiv I used here? Would the sentence have another meaning if "sind" were used instead?

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No one claims here, that such functions exist. Therefore you can't use "sind" here. – bernd_k Jul 6 '11 at 10:08
"A sei B" ist gebräuchliche mathematische Ausdrucksweise. Statt zu sagen, die Größe A soll den Wert B haben, wird kürzer "A sei B" gesagt. In Grammatiken werden spezielle Ausdrucksweisen von Fachsprachen normalerweise nicht erfaßt, aber dieser Konjunktiv stimmt mit dem Anwendungsbereich Wunsch, Forderung überein. – rogermue Nov 2 '14 at 15:24

6 Answers 6

It is the old function of the optative which the Konjunktiv 1 has lost as productive feature between 1000 and 1500 AD in German (coniunctivus optativus). It's preserved in recipes, proverbs and some phrases:

Man nehme ein Ei und schlage es sich kräftig gegen den Kopf.

It is preserved in other languages, too:

God save the Queen.
Vive la France.
Requiescat in pace! – Er ruhe in Frieden!

The relict use is restricted to the third person. All other forms of sei- are real imperative forms, so this sei- of the third person could be set into imperative form paradigm, but the problem is that this use is strictly limited to the sound of recipes and, of course, mathematics. It has a long tradition there, because the optativus is used in logics and philosophy for a very long time.

We (Belles Lettres) have an article and a video-tutorial about this (in German).

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+1 for this excellent answer (and the recipe "Man nehme ein Ei und schlage es sich kräftig gegen den Kopf.") – splattne Jul 6 '11 at 14:20
And kudos on your site! That link is just excellent. :) – Stovner Jul 8 '11 at 8:25
This answer needs improvement: In mathematics, it is even more common to put the verb in the first position of the sentence, e.g. „Seien A, B Mengen.“ – now, why is this possible? I know that an imperative takes the first position in a sentence – can a subjunctive as well? – k.stm Oct 9 '12 at 8:36

No, that's not subjunctive mood (Konjunktiv), it's the imperative mood.

When in a mathematical proof you postulate something, the imperative form is used to 'bring something into existence'. Just like

Es werde Licht!

EDIT: 4. Definieren mit Konditionalgefüge describes this form as an ancient form of imperative, and compares it with the Prussian kings signing their orders with "Es sei!"

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I had never heard of third-person imperative. How to construct it? Do you have any support for that statement? Your link doesn't mention imperative. – Tim Jul 6 '11 at 10:04
@TimN: The example is from the bible, and it is third-person imperative ("Let there be light!"). Another common expression is "So sei es!", meaning "So soll es sein!". It seems that technically, they're both Konjunktiv, but it is not indirect speech. – OregonGhost Jul 6 '11 at 10:28
Firstly, the links are outdated. Secondly: As far as I know, if an imperative is used, syntactically the verb in imperative takes the first position in a sentence, e.g.: „Öffne er die Türe!“. Also, exclamation marks are used to indicate imperative moods, why is it that they are missing and the verb takes the second position (it can take the first, e.g. „Seien A, B Mengen.“)? Also, according to the answer by Belles Lettres, this is wrong. – k.stm Oct 9 '12 at 8:30

In a mathematical context, the Konjunktiv has the sense of "suppose" or "let".

Die Funktionen ψn(x) seien Eigenfunktionen eines Hamiltonoperators Ĥ


Let the functions ψn(x) be the eigenfunctions of a Hamiltonian operator Ĥ.

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I think it is because you "define" what the function is. For example, in your sentence you define "ψn(x)" to be "Eigenfunktionen eines Hamiltonoperators Ĥ.", which they were not until you said so, so you cannot use "sind".

Like @splattne correctly said, you can prepend "Angenommen, ..." ("Assuming...") to the statement to make it more clear.

A reference to using it that way in mathematics:

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I agree. For the same reason, you say "let x be an element of X" instead of "x is an element of X" in English texts. – Stefan Walter Jul 6 '11 at 10:06
The linked article explains that the form in the bible "Es werde Licht!" and the one in mathematics are the same and both Konjunktiv. This is true; but the important aspect is that it is third-person imperative, not indirect speech. The "Angenommen" prefix changes the meaning: With it, it is no longer imperative and no longer defines something, it just assumes something. Haarspalterei. For mathematics, the difference does not matter. – OregonGhost Jul 6 '11 at 10:33
@OregonGhost: a "third-person imperative" in German? This is new to me. Do you have a source for it? – splattne Jul 6 '11 at 10:56
@splattne: It is archaic today and seems not to be regulated in the Amtliche Rechtschreibung. You'll find it in Goethe's work ("Geh er mir aus dem Weg!") and in the bible. Note, however, that in any case, the form is basically identical to Konjunktiv I, which is why it does not need to be differentiated today. I've only found one two-part source in English so far (read both articles, the parts about third-person commands). – OregonGhost Jul 6 '11 at 11:42
"In altertümlichen Texten auch für den Imperativ in der dritten Person: "Gott sprach: 'Es WERDE Licht!'", oder [...] "Ich sei, gewährt mir die Bitte..." (das also keine Form der indirekten Rede, sondern ein Imperativ ist, der sich an eine unbestimmte dritte Größe richtet) So was gibts im normalen Gesprächsleben heute nicht mehr." - Unfortunately, there's also no source given. – OregonGhost Jul 6 '11 at 12:00

The exact same thing happens in English. One would say:

Let ψn(x) be the characteristic function of the Hamiltonian operator Ĥ.

This has a different meaning than:

ψn(x) is the characteristic function of the Hamiltonian operator Ĥ.

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This should be a comment (which I know you cannot post yet). – Wrzlprmft Oct 3 '14 at 15:32

In mathematics you often set up a situation with this kind of phrase - often to express the logical link: If ... then .... In German, using the example of the question it looks like this:

Die Funktionen ψn(x) seien Eigenfunktionen eines Hamiltonoperators Ĥ. Dann gilt ...

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