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Unformatted text preview: Linear Vector Spaces in Quantum Mechanics We have observed that most operators in quantum mechanics are linear operators. This is fortunate because it allows us to represent quantum mechanical operators as matrices and wavefunctions as vectors in some linear vector space. Since computers are particularly good at performing operations common in linear algebra (multiplication of a matrix times a vector, etc.), this is quite advantageous from a practical standpoint. In an -dimensional space we may expand any vector as a linear combination of basis vectors (80) For a general vector space, the coefficients may be complex; thus one should not be too quick to draw parallels to the expansion of vectors in three-dimensional Euclidean space. The coefficients are referred to as the ``components'' of the state vector , and for a given basis, the components of a vector specify it completely. The components of the sum of two vectors are the sums of the components. If and then (81) and similarly (82) The scalar product of two vectors is a complex number denoted by (83) where we have used the standard linear-algebra notation. If we also require that (84) then it follows that...
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This note was uploaded on 11/22/2011 for the course CHEMISTRY CHM1025 taught by Professor Laurachoudry during the Fall '10 term at Broward College.
- Fall '10