David Blei has a great talk introducing LDA to students of a summer class: http://videolectures.net/mlss09uk_blei_tm/
In the first video he covers extensively the basic idea of topic modelling and how Dirichlet distribution come into play. The plate notation is explained as if all hidden variables are observed to show the dependencies. Basically topics are distributions over words and document distributions over topics.
In the second video he shows the effect of alpha with some sample graphs. The smaller alpha the more sparse the distribution. Also, he introduces some inference approaches.
Can LDA be used to detect the topic of A SINGLE document?
Yes, in its particular representation of 'topic,' and given a training corpus of (usually related) documents.
LDA represents topics as distributions over words, and documents as distributions over topics. That is, one very purpose of LDA is to arrive at probabilistic representation of each document as a set of topics. For example, the LDA implementation in gensim can return this representation for any given document.
But this depends on the other documents in the corpus: Any given document will have a different representation if analyzed as part of a different corpus.
That's not typically considered a shortcoming: Most applications of LDA focus on related documents. The paper introducing LDA applies it to two corpora, one of Associated Press articles and one of scientific article abstracts. Edwin Chen's nicely approachable blog post applies LDA to a tranche of emails from Sarah Palin's time as Alaska governor.
If your application demands separating documents into known, mutually exclusive classes, then LDA-derived topics can be used as features for classification. Indeed, the initial paper does just that with the AP corpus, with good results.
Relatedly, Chen's demonstration doesn't sort documents into exclusive classes, but his documents' mostly concentrate their probability on single LDA topics. As David Blei explains in this video lecture, the Dirichlet priors can be chosen to favor sparsity. More simply, "a document is penalized for using many topics," as his slides put it. This seems the closest LDA can get to a single, unsupervised topic, but certainly doesn't guarantee every document will be represented as such.
Best Answer
You can use a sentence splitter and split your document into sentences. I have never used the approach myself, but the tool is available with the open.nlp package in R, Python and Rapidminer.
What you could also do is to train a topicmodel on corpus with clearly defined topics. Next you use the same model on your one document and you see how the topic structure turn out.