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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 “braket” 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 = cψ1 ψ2 cψ1 ψ2 = c∗ ψ1 ψ2 ψ1 and ψ2 are orthogonal if ψ1 ψ2 = 0
The normalization condition is drψ (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 <ψψ>. 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, nm = 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 = nψ 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 nn = 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 <vu> = <uv>*
13 C. Basis Sets C.5 Matrix Representation of Linear Operators Ωv = 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 Ωv1 + λ2 Ωv2 14 C. Basis Sets C.5 Matrix Representation of Linear Operators Ωv = 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 = nΩm = 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 = nΩ† m ≡ Ωnm
∗ 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 selfadjoint. Selfadjoint operators are called Hermitian. ψ1 Ωψ2 = 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”) Ωω = 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 αα = 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
nm = δnm drϕ∗ (r)ϕm (r) = δnm
n nn = 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 αβ = δ (α − β ) ∗
drωα (r)ωβ (r) = δ (α − β ) dααα = 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) = αψ 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) dαc(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 αβ = δ (α − β ) ∗
drωα (r)ωβ (r) = δ (α − β ) dααα = 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). rψ = 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 braket 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 braket 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. rX ψ = xrψ
• 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. rY ψ = y rψ
rZ ψ = z rψ 26 F. The Position and Momentum Operators
F.1 The Position Operator rX ψ = xrψ
rY ψ = y rψ
rZ ψ = z rψ • The X, Y and Z operators can be combined into one vector operator R. rRψ = rrψ
• 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. pPx ψ = px pψ • We can similarly write Py and Pz momentum operators. pPy ψ = py pψ
pPz ψ = pz pψ • The Px, Py and Pz operators can be combined into one vector operator P. pPψ = ppψ
• This deﬁnition means that in the {r>} basis we get
rPψ = ∇rψ
i
28 F. The Position and Momentum Operators
F.2 The Momentum Operator • This deﬁnition means that in the {r>} basis we get
rPψ = ∇rψ
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 ] = iδij 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 ...
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This note was uploaded on 09/28/2011 for the course PHYS 334 taught by Professor Resch during the Spring '08 term at Waterloo.
 Spring '08
 RESCH
 mechanics

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