Quantum mechanics is deeply connected with representation theory. Therefore, I'm looking for a textbook or article which presents quantum mechanics in a representation theoretic manner. Could anyone provide me a reference to such a resource? I don't care if it's a math or physics book as long as it makes use of the language of representation theory mathematicians are familiar with.

6$\begingroup$ Peter Woit wrote some good lecture notes in this spirit based on some advanced undergraduate classes that he taught last year: math.columbia.edu/~woit/QM/fall2012.html and math.columbia.edu/~woit/QM/spring2013.html $\endgroup$ – Paul Siegel Jan 22 '14 at 21:24

1$\begingroup$ faculty.virginia.edu/quantum/MathSeminar2003_04_23.pdf $\endgroup$ – Carlo Beenakker Jan 22 '14 at 21:57

$\begingroup$ Maybe this should be Community Wiki? $\endgroup$ – Tobias Fritz Jan 29 '14 at 10:07
Here are some references.
 Hermann Weyl, The Theory of Groups and Quantum Mechanics, Dover Publications, 1950
 George W. Mackey, Induced representations of groups and quantum mechanics, W. A. Benjamin, 1968
 M. Gourdin , Unitary Symmetries, John Wiley & Sons, New York, 1968.
 A.R. Edmonds, Angular Momentum in Quantum Mechanics, Princeton, 1968
 L. C. Biedenharn, James D. Louck, Angular Momentum in Quantum Physics: Theory and Application, Cambridge University Press, 2009
 Gianni Cassinelli, The Theory of Symmetry Actions in Quantum Mechanics: With an Application to the Galilei Group, Springer, 2004
I believe 1 and 2 are classics, 3 and 4 probably nearly so. I'm not familiar with 5 nor 6 in detail, but they look interesting  I'm very curious to see comments by other users on any of the above.
Basic foundations of quantum mechanics in a representation theoretic language can be found in van den Ban's lectures "Applications of representation theory in classical quantum mechanics" at this page http://www.staff.science.uu.nl/~ban00101/lecnot.html
The following articles can be also relevant: http://link.springer.com/article/10.1023%2FB%3AACAP.0000024199.14167.44 (N. P. Landsman, Quantum Mechanics and Representation Theory: The New Synthesis). http://arxiv.org/abs/0711.2699 (I. Frenkel, M. Libine, Quaternionic Analysis, Representation Theory and Physics)
There is an upcoming book of Peter Woit, available for download if you click on the link. From the Preface:
These are the course notes prepared for a class taught at Columbia during the 201213 academic year. The intent was to cover the basics of quantum mechanics, from a point of view emphasizing the role of unitary representations of Lie groups in the foundations of the subject. [..]
I just noticed that these lecture notes were also mentioned in the comments.