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Unformatted text preview: Physical Chemistry Course Number: C362 1 The mathematical ideas you will encounter while learning quantum mechanics We first outline a few mathematical ideas that one needs to tackle quantum me- chanics. This section includes a brief review of: Vectors and matrices Differentiation and integration Complex numbers 1.1 Vectors and matrices 1. We will start with simple concepts from three-dimensional vector spaces. 2. In 3D, a vector has three components: vector r = x i + y j + z k. (1.1) Here, i , j and k are unit vectors in the x, y and z directions respectively. i , j and k may also be called basis vectors or just bases and this is a terminology that we will use often. Vectors can be represented in the following form: vector r x y z (1.2) 3. A vector is fully described by: (i) its magnitude, and (ii) its direction. Com- pare this with a scalar number that only has magnitude. 4. The magnitude of the vector vector r can be calculated by using the concept of a dot product: | vector r | vector r vector r = radicalBig | x | 2 + | y | 2 + | z | 2 (1.3) where we have introduced the definition of the dot product of two vectors: vectora = a 1 i + a 2 j + a 3 k vector b = b 1 i + b 2 j + b 3 k vectora vector b = a 1 b 1 + a 2 b 2 + a 3 b 3 (1.4) Chemistry, Indiana University 4 c circlecopyrt 2011, Srinivasan S. Iyengar (instructor) Physical Chemistry Course Number: C362 5. The dot product is a very fundamental concept, and very useful for our further development. Physically a dot product of vectora with vector b is equal to the magnitude of vectora times the projection of vector b onto vectora . See figure on board....
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- Winter '11