Ch2 - Formalism

Ch2 - Formalism - Chapter 2 Formalism of Quantum Mechanics...

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Formalism of Quantum Mechanics Chapter 2 Bransden and Joachain Chapter 5 Sakurai Chapter 1,2 Shankar Chapter 1
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00 () x e ϕ SHO energy eigenfunctions 11 x e 22 x e 33 x e 44 x e 55 x e By completeness and orthogonality of energy eigenfunctions, any wave function can be uniquely expanded in a linear superposition: Ket vector representation Treating the wave function as a vector in a linear complex vector space H . 0 nn n x cx = Ψ= 0 n ce =
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Linear Vector Space (Basics) Definition: A linear vector space V is a collection of objects called vectors, for which there exists (1) A definite rule for forming vector sum (2) A definite rule for multiplication for scalars a, 1 , 2 ,. .., , ,... vw 12 + 1 a •Closure: • Distributive: • Commutative: • Associative: • Existence of a unique null vector: • Existence of inverse under addition: uv V +∈ () ( ) , au v a u a v a bu a u b u += + + = + ( ) wu v w u v ++= ++ vu +=+ u null u + = u u null + −= 0 u null =
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Linear Vector Space (Basics) Definition: The set of vectors is said to be linearly independent if and only if all for the following equation 1 , 2 ,. .., n 1 n i i a i null = = 0 i a = Theorem: Any vector in an n -dimensional space can be written as a linear combination of n independent vectors 1 , 2 ,. .., n 1 .. , n i i ie v c i = = Definition: A set of vectors n linearly independent vectors in an n -dimensional space is called basis Unique expansion
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Linear Vector Space (Basics) For any two vectors the inner product is denoted by , which is a number (generally complex) dependent on the two vectors, satisfying the following rules: , uv •Skew-symmetry • Positive semi-definiteness • Linearity * vu = 0 vv equals zero iff v null = ( ) ua v b w a u v b u w += + Definition: A vector space with an inner product is called an inner product space Definition: Two vectors are said to be orthogonal if their inner product is zero Definition: The norm of the vector is defined by v Definition: A set of basis vectors all of unit norm, and are pairwise orthogonal, is called an orthonormal basis
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Postulates in Quantum Mechanics (I) The state of a given physical system at any instant t is defined by a ket , which is termed the quantum state vector belonging to a linear complex vector space called Hilbert space H . The time evolution of is governed by () t ψ (IIa) Any measurable physical quantity (observable) is described by a corresponding Hermitian operator A acting in H. The measurement of the physical quantity can only yield one of the eigenvalues of the corresponding operator, (IIb) The probability of obtaining the eigenvalue associated with a non-degenerate eigenvector is given by n n λ n A nn = n (IIc) After an ideal measurement of A , the state vector collapses into an eigenvector of A corresponding to the measurement outcome of A . tn t Schrödinger equation t Ht i t = = 2 |( ) | Pnt =
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Expectation value of observables M is hermitian { } n is complete and orthonormal Probability of the system in n • Expectation value of M = n M nn λ = An observable M n n cn Ψ= 2 2 || n = Ψ 2 n cM M Representation of the system state vector in the measurement basis M
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Ch2 - Formalism - Chapter 2 Formalism of Quantum Mechanics...

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