Lecture8Notes - Chem 120A 02/03/06 Spring 2006 READING: To...

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Chem 120A Dirac Notation and Linear Operators 02/03/06 Spring 2006 Lecture 8 READING: To be announced Rules of quantum mechanics In the last lecture, we examined the Dirac notation and how it is used in quantum mechanics. The following rules (a.k.a. postulates) of quantum mechanics were presented: • Rule 1 : Quantum mechanical states can be expressed as kets or bras. Bras are the complex conjugates of kets. Both bras and kets reside in complex vector spaces. • Rule 2 : A state expressed as a bra or ket can be fully described by a superposition within a basis that spans the vector space. In order to express the state ± ± s ² as a superposition in the basis { α i } , we must know all of the superposition coefficients, ³ i ± ± s ² Here, we will consider the properties of the superposition coefficients. A state ± ± s ² expressed in the { i } basis as ± ± s ² = N i = 1 ³ i ± ± s ² ± ± i ² = N i = 1 A i ± ± i ² , (1) where the { A i } are the superposition coefficients. Similarly, a state ³ s ± ± can be expressed in the { i } basis as ³ s ± ± = N i = 1 ³ s ± ± i ²³ i ± ± = N i = 1 B i ³ i ± ± , (2) where the { B i } are the superposition coefficients. What is the relation between A i and B i ? To examine this, Chem 120A, Spring 2006, Lecture 8 1
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we will consider the inner products ± s ² ² α j ³ and ± j ² ² s ³ , where j is a single vector from the basis { i } : ± j ² ² s ³ = i A i ± j ² ² i ³ = i A i δ ij = A j . (3) This first inner product reduces to a single superposition coefficient, A j , because the Kronecker delta is 0 unless i = j . Similarly, the second inner product also reduces to a single coefficient: ± s ² ² j ³ =
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Lecture8Notes - Chem 120A 02/03/06 Spring 2006 READING: To...

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