2_Formalism

2_Formalism - II. The Formalism of Quantum Mechanics...

Info iconThis preview shows page 1. Sign up to view the full content.

View Full Document Right Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: II. The Formalism of Quantum Mechanics Griffiths chapter 3 1 A. Dirac Notation ￿ψ1 |ψ2 ￿ ≡ • Less writing. ￿ (Griffiths 3.6) ∗ drψ1 (r)ψ2 (r) (single particle case) • Makes mathematical manipulations easier. • Independent of basis. ￿ψ1 |ψ2 ￿ ≡ ￿ Bars denote Fourier transforms of ψ. ¯∗ (p)ψ2 (p) ¯ dpψ1 (momentum space) 2 A. Dirac Notation ￿ψ1 |ψ2 ￿ ≡ ￿ ∗ drψ1 (r)ψ2 (r) • |ψ2> : “ket”. It is a vector. (More later.) • <ψ1| : “bra”. It is a linear function of vectors. For example: ￿ψ1 | = ￿ ∗ drψ1 (r)[· · · ] • It will inevitably operate on a ket to give a complex number (inner product). • Thus Dirac notation is sometimes called “bra-ket” notation. 3 A. Dirac Notation Some properties: associative Let c be a complex number. commutative ￿ψ1 |ψ2 ￿ = ￿ψ2 |ψ1 ￿∗ ￿ψ3 |ψ1 + ψ2 ￿ = ￿ψ3 |ψ1 ￿ + ￿ψ3 |ψ2 ￿ distributive ￿ψ1 |cψ2 ￿ = c￿ψ1 |ψ2 ￿ ￿cψ1 |ψ2 ￿ = c∗ ￿ψ1 |ψ2 ￿ ψ1 and ψ2 are orthogonal if ￿ψ1 |ψ2 ￿ = 0 The normalization condition is ￿ ￿ dr|ψ (r)|2 = 1 If the functions are both orthogonal and normalized, then they are orthonormal. drψ ∗ (r)ψ (r) = 1 ￿ψ |ψ ￿ = 1 4 B. Hilbert Space (Griffiths 3.1) • A vector exists geometrically, independent of a coordinate system. v • We can represent it in terms of a basis set, (êx, êy, êz), for example, where (êx, êy, êz) are unit vectors that span the space. z ˆ ez x ˆ ex ˆ ˆ ˆ v = vx ex + vy ey + vz ez v ˆ ey z’ y v ˆ ez x’ ￿ ￿ ￿ ˆ ˆ ˆ v = vx ex￿ + vy ey￿ + vz ez￿ ’ ˆ ex ’ ˆ ey ’ y’ 5 B. Hilbert Space • The inner product of two vectors is a scalar. v · u = vx ux + vy uy + vz uz ￿ ￿ ￿ v · u = vx u￿ + vy u￿ + vz u￿ x y z • The length of the vector is √(v⋅v). • A Hilbert space is the same type of space except it is for functions instead of discrete element (traditional) vectors. f(x) f(x3) f(x2) f(x1) f(xn) Δ x1 x2 x3 f = [f (x1 ), f (x2 ), · · · f (xn )] with respect to the basis (x1 , x2 , · · · , xn ) xn x v = (vx , vy , vz ) Compare with with respect to the basis ˆˆˆ (ex , ey , ez ) 6 B. Hilbert Space f(x) f(x3) f(x2) f(x1) f(xn) Δ x1 x2 x3 f = [f (x1 ), f (x2 ), · · · f (xn )] with respect to the basis (x1 , x2 , · · · , xn ) xn x v = (vx , vy , vz ) Compare with with respect to the basis ˆˆˆ (ex , ey , ez ) • The limit n → ∞ while the spacing Δ → 0 gives us the state function (wave function). • These are sometime called state vectors. • Functions are vectors in Hilbert space (a vector space in infinite dimensions). 7 B. Hilbert Space Some properties: 1. The space is linear. - If ψ ∈ space and c is a constant then cψ ∈ space. - If ψ1, ψ2 ∈ space then ψ1+ψ2 ∈ space. 2. There is an inner product. If ψ1, ψ2 ∈ space then ￿ψ1 |ψ2 ￿ ≡ ￿ ays lw ea e w h y t wa v o b e w t i s i s y t h a h ave ble”. h sa T a s i o n i n te g r ct f u n u a re “sq ∗ drψ1 (r)ψ2 (r) exists. 3. The length of a vector is related to the inner product <ψ|ψ>. Compare with the “ length” of a “regular” vector, v⋅v. C om p a r e with “re g u l a r ” ve c t o r p ro duc t i n ne r v·u= vx ux + . vy uy + vz uz 8 C. Basis Sets (Griffiths 3.6 and appendix A) • Just as there are sets of vectors that span “regular” vector space – for example (êx, êy, êz) – there can be functions that span Hilbert space. {ϕ1 (r), ϕ2 (r), · · · } → {ϕn (r)} or, taking some liberties with the n otation as will be explained later, we can write {|ϕ1 ￿, |ϕ2 ￿, · · · } → {|ϕn ￿} or just {|1￿, |2￿, · · · } → {|n￿} 9 C. Basis Sets C.1 Orthonormality Compare ˆˆ ei · ej = δij for i,j = x, y or z, with ￿ drϕ∗ (r)ϕm (r) n orthonormal basis set (êx, êy, êz) f or a “regular” vector space = δnm orthonormal basis set {|n>} f or a Hilbert space or, in Dirac notation, ￿n|m￿ = δnm Kronecker delta δnm = 1 i f n = m = 0 if n ≠ m 10 C. Basis Sets C.2 Projections Compare ˆ ˆ ˆ v = vx ex + vy ey + vz ez v= 3 ￿ y v ˆ vn en n=1 with ψ (r) = vx ￿ vy x cn ϕn (r) n or |ψ ￿ = ￿ n cn |n￿ • The coefficients cn are projections of ψ onto the vectors |n>. cn = ￿n|ψ ￿ See board – II.C.2 11 C. Basis Sets C.3 The Closure Relation • An orthonormal set {|n>} is a basis set if, for every function ψ(r), the function can be expressed as an expansion in {|n>}. ￿ n ϕ∗ (r￿ )ϕn (r) = δ (r − r￿ ) n Di ra See c de l bo t a f u n ard c t i – I on I.C . In Dirac notation: ￿ n Closure Relation |n￿￿n| = I 3.a See board – II.C.3.b Ide nti ty o perato r • The closure relation must be satisfied if the set {|n>} is a basis set. 12 C. Basis Sets C.4 Matrix Representation of Vectors • The inner product of wave functions can be written more simply when the wave functions are expressed in terms of a basis set. ￿v |u￿ = or in matrix notation ￿ n ∗ vn un ∗∗ ∗ ￿v |u￿ = [v1 , v2 , · · · , vn ] u1 u2 . . . un See board – II.C.4 • With respect to a basis set, the bras and kets are identified with row and column matrices. ￿v | → ke to ta ow s e e h r a k e t. an , we c f a bra o his rom t dj o i nt o F the a ∗∗ [v1 , v2 , · · · |u￿ → u1 u2 . . . un ∗ , vn ] Re m emb e con r the c jug a om te s ! p le x <v|u> = <u|v>* 13 C. Basis Sets C.5 Matrix Representation of Linear Operators Ω|v ￿ = |v ￿ ￿ operator (Could be a derivative, L aplacian, Identity, etc.) wave function (state vector) new wave function (state vector) Definition of a Linear Operator Ω [λ1 |v1 ￿ + λ2 |v2 ￿] = λ1 Ω|v1 ￿ + λ2 Ω|v2 ￿ 14 C. Basis Sets C.5 Matrix Representation of Linear Operators Ω|v ￿ = |v ￿ ￿ • With respect to an orthonormal basis {|n>} this can be written as ￿ ￿ Ωnm vm = vn m or in matrix notation where Ω11 Ω21 . . . Ω n1 Ω12 Ω22 ··· ··· .. . Ω n2 ··· See board – II.C.5.a Ω1 n Ω2 n Ωnn Ωnm = ￿n|Ω|m￿ = ￿ v1 v2 . . . vn ￿ v1 ￿ v2 = . . . ￿ vn drϕ∗ (r)Ωϕm (r) n 15 C. Basis Sets C.5 Matrix Representation of Linear Operators • From linear algebra, the adjoint of a matrix is the transpose and complex conjugate of that matrix. complex conjugate † Ωnm = ∗ Ωmn transpose • The adjoint of an operator, Ω✝, means that the operator Ω is associated with the bra instead of the ket. = drϕ∗ (r)Ω† ϕm (r) n ￿ = ￿ ￿n|Ω† |m￿ ≡ ￿Ωn|m￿ ∗ dr [Ωϕn (r)] ϕm (r) See board – II.C.5.b 16 C. Basis Sets C.5 Matrix Representation of Linear Operators Some properties: 1. The adjoint of a product is † (ΩΛ) = Λ† Ω† 2. If Ω✝ = Ω, it is self-adjoint. Self-adjoint operators are called Hermitian. ￿ψ1 |Ω|ψ2 ￿ = ￿Ωψ1 |ψ2 ￿ • Hermitian operators are important because they are related to observables in quantum mechanics. • See Griffiths 3.2.1. 17 D. The Eigenvalue Problem (Griffiths 3.2, 3.3 and A.5) (Characteristic values, or from the German eigenwerte → “proper value”) Ω|ω ￿ = ω |ω ￿ operator vector (eigenvector of Ω) Eigenvalue equation (Eigenvalue problem) number (eigenvalue of Ω) D.1 The Eigenvalues of a Hermitian Operator Are Real Proof: See board – II.D.1 18 D. The Eigenvalue Problem D.2 The Eigenfunctions of a Hermitian Operator Are Orthogonal Proof: See board – II.D.2 If ￿ α |α￿￿α| = I where {|α>} are the eigenfunctions of a Hermitian operator, then that operator is defined as an observable. • Not all Hermitian operators are observables. 19 E. Generalization to Continuous Basis Sets (Griffiths 3.3) • A discrete, orthonormal, basis set {|n>} can be defined as orthonormal ￿ ￿n|m￿ = δnm drϕ∗ (r)ϕm (r) = δnm n ￿ |n￿￿n| = I ￿n ϕ∗ (r￿ )ϕn (r) n n spans the space = δ (r − r￿ ) • Let’s define a continuous, orthonormal, basis set as a set of functions {ωα(r)}, where α is a continuous index, such that ￿ orthonormal ￿ ￿α|β ￿ = δ (α − β ) ∗ drωα (r)ωβ (r) = δ (α − β ) ￿ dα|α￿￿α| = I spans the space ∗ dαωα (r￿ )ωα (r) = δ (r − r￿ ) 20 E. Generalization to Continuous Basis Sets E.1 Continuous Components of a Wavefunction • General wave functions can be expanded in terms of continuous basis sets. ￿ ￿ ψ (r) = or in Dirac notation |ψ ￿ = ￿ dαc(α)ωα (r) where where dαc(α)|α￿ c(α) = ∗ drωα (r)ψ (r) c(α) = ￿α|ψ ￿ See board – II.E.1 E.2 Scalar Product and Norm in Terms of Continuous Components • Assume ψ1(r) and ψ2(r) have continuous components b(α) and c(α), respectively, with respect to the continuous basis set {|α>}. • The inner product between them will be ￿ψ1 |ψ2 ￿ = • From this, the norm will be ￿ψ |ψ ￿ = ￿ ￿ dαb∗ (α)c(α) dα|c(α)|2 See board – II.E.2 21 E. Generalization to Continuous Basis Sets E.3 The Position Representation • Let us introduce a continuous basis set ξr’(r) defined as ξr￿ (r) = δ (r − r￿ ) continuous index r’ The index r’ is also a vector. (Three continuous indices.) • We can abbreviate this basis set with the usual Dirac notation. ξr￿ (r) → |r ￿ ￿ • This basis set meets the criteria for a continuous basis set. 22 E. Generalization to Continuous Basis Sets E.3 The Position Representation • This basis set meets the criteria for a continuous basis set. orthonormal ￿ ￿α|β ￿ = δ (α − β ) ∗ drωα (r)ωβ (r) = δ (α − β ) ￿ ￿ dα|α￿￿α| = I spans the space ∗ dαωα (r￿ )ωα (r) = δ (r − r￿ ) See board – II.E.3.a • Using this position basis set, we see that any wave vector |ψ> represented with respect to it is written as ψ(r). ￿r|ψ ￿ = ψ (r) basis set See board – II.E.3.b vector components (projections) o f the vector 23 E. Generalization to Continuous Basis Sets E.3 The Position Representation • Thus we get back our bra-ket representation of the Hilbert space inner product that was our original definition. ￿ψ1 |ψ2 ￿ ≡ ￿ ∗ drψ1 (r)ψ2 (r) See board – II.E.3.c E.4 The Momentum Representation • We can also define a continuous basis set that gives us the momentum representation. −3 2 vp￿ (r) = (2π ￿) e i￿ ￿ p ·r continuous index p’ The index p’ is also a vector. (Three continuous indices.) 24 E. Generalization to Continuous Basis Sets E.4 The Momentum Representation • We can also define a continuous basis set that gives us the momentum representation. −3 2 vp￿ (r) = (2π ￿) e i￿ ￿ p ·r • We can abbreviate this basis set with the usual Dirac notation. vp￿ (r) → |p￿ ￿ • Thus we get back, again, the bra-ket representation of the Hilbert space inner product that was our alternative definition. ￿ψ1 |ψ2 ￿ ≡ ￿ ¯∗ ¯ dpψ1 (p)ψ2 (p) See board – II.E.4 25 F. The Position and Momentum Operators (Griffiths 3.3) F.1 The Position Operator • The position operator X is defined to be the operator that, in the {|r>} representation, multiplies by the index x. ￿r|X |ψ ￿ = x￿r|ψ ￿ • This can be written more intuitively as X ψ (r) = xψ (r) • So the X position operator has the position x of a quantum mechanical particle as eigenvalues. • We can similarly write Y and Z position operators. ￿r|Y |ψ ￿ = y ￿r|ψ ￿ ￿r|Z |ψ ￿ = z ￿r|ψ ￿ 26 F. The Position and Momentum Operators F.1 The Position Operator ￿r|X |ψ ￿ = x￿r|ψ ￿ ￿r|Y |ψ ￿ = y ￿r|ψ ￿ ￿r|Z |ψ ￿ = z ￿r|ψ ￿ • The X, Y and Z operators can be combined into one vector operator R. ￿r|R|ψ ￿ = r￿r|ψ ￿ • These operators give the expectation values of the position components of a particle. For example, <X> gives ￿X ￿ = ￿ drψ ∗ (r)xψ (r) See board – II.F.1 27 F. The Position and Momentum Operators F.2 The Momentum Operator • The momentum operator Px is defined to be the operator that, in the {|p>} representation, multiplies by the index px. ￿p|Px |ψ ￿ = px ￿p|ψ ￿ • We can similarly write Py and Pz momentum operators. ￿p|Py |ψ ￿ = py ￿p|ψ ￿ ￿p|Pz |ψ ￿ = pz ￿p|ψ ￿ • The Px, Py and Pz operators can be combined into one vector operator P. ￿p|P|ψ ￿ = p￿p|ψ ￿ • This definition means that in the {|r>} basis we get ￿ ￿r|P|ψ ￿ = ∇￿r|ψ ￿ i 28 F. The Position and Momentum Operators F.2 The Momentum Operator • This definition means that in the {|r>} basis we get ￿ ￿r|P|ψ ￿ = ∇￿r|ψ ￿ i • This can be written in a more familiar way as ￿ ∂ψ (r) Px ψ (r) = i ∂x ￿ ∂ψ (r) Py ψ (r) = i ∂y ￿ ∂ψ (r) Pz ψ (r) = i ∂z • The expectation value of Px is ￿Px ￿ = ￿ ￿ ∂ψ (r) drψ (r) i ∂x • Py and Pz expectation values are similar. See board – II.F.2 29 F. The Position and Momentum Operators F.3 R and P Commutation Relations and the Uncertainty Principle • Two operators A and B commute if AB = BA. • The commutator of A and B is defined as [A,B] = AB - BA. • Therefore the operators commute if the commutator is zero. That is, if [A,B] = 0. • The commutator between R and P is [Ri , Pj ] = i￿δij where i, j = x, y, z See board – II.F.3.a • From this commutation relation we get the Heisenberg Uncertainty Principle: ￿ ∆ px ∆ x ≥ 2 • Similarly, ΔpyΔy ≥ ħ/2 and ΔpzΔz ≥ ħ/2. See board – II.F.3.b 30 31 ...
View Full Document

This note was uploaded on 09/28/2011 for the course PHYS 334 taught by Professor Resch during the Spring '08 term at Waterloo.

Ask a homework question - tutors are online