Quantum Mechanics Theorems with Proof

Part IB — Quantum Mechanics Theorems with proof

Based on lectures by J. M. Evans Notes taken by Dexter Chua

Michaelmas 2015

These notes are not endorsed by the lecturers, and I have modiﬁed them (often signiﬁcantly) after lectures. They are nowhere near accurate representations of what was actually lectured, and in particular, all errors are almost surely mine.

Physical background Photoelectric eﬀect. Electrons in atoms and line spectra. Particle diﬀraction. [1]

Schr¨odingerequation and solutions De Broglie waves. Schr¨odinger equation. Superposition principle. Probability interpre- tation, density and current. [2] Stationary states. Free particle, Gaussian wave packet. Motion in 1-dimensional potentials, parity. Potential step, square well and barrier. Harmonic oscillator. [4]

Observables and expectation values Position and momentum operators and expectation values. Canonical commutation relations. Uncertainty principle. [2] Observables and Hermitian operators. Eigenvalues and eigenfunctions. Formula for expectation value. [2]

Hydrogen atom Spherically symmetric wave functions for spherical well and hydrogen atom. Orbital angular momentum operators. General solution of hydrogen atom. [5]

1 Contents IB Quantum Mechanics (Theorems with proof)

Contents

0 Introduction 3 0.1 Light quanta ...... 3 0.2 Bohr model of the atom ...... 3 0.3 Matter waves ...... 3

1 Wavefunctions and the Schr¨odingerequation 4 1.1 Particle state and probability ...... 4 1.2 Operators ...... 4 1.3 Time evolution of wavefunctions ...... 4

2 Some examples in one dimension 5 2.1 Introduction ...... 5 2.2 Inﬁnite well — particle in a box ...... 5 2.3 Parity ...... 5 2.4 Potential well ...... 5 2.5 The harmonic oscillator ...... 5

3 Expectation and uncertainty 6 3.1 Inner products and expectation values ...... 6 3.2 Ehrenfest’s theorem ...... 7 3.3 Heisenberg’s uncertainty principle ...... 7

4 More results in one dimensions 9 4.1 Gaussian wavepackets ...... 9 4.2 Scattering ...... 9 4.3 Potential step ...... 9 4.4 Potential barrier ...... 9 4.5 General features of stationary states ...... 9

5 Axioms for quantum mechanics 10 5.1 States and observables ...... 10 5.2 Measurements ...... 10 5.3 Evolution in time ...... 11 5.4 Discrete and continuous spectra ...... 11 5.5 Degeneracy and simultaneous measurements ...... 11

6 Quantum mechanics in three dimensions 12 6.1 Introduction ...... 12 6.2 Separable eigenstate solutions ...... 12 6.3 Angular momentum ...... 12 6.4 Joint eigenstates for a spherically symmetric potential ...... 13

7 The hydrogen atom 14 7.1 Introduction ...... 14 7.2 General solution ...... 14 7.3 Comments ...... 14

2 0 Introduction IB Quantum Mechanics (Theorems with proof)

0 Introduction 0.1 Light quanta 0.2 Bohr model of the atom 0.3 Matter waves

3 1 Wavefunctions and the Schr¨odingerequationIB Quantum Mechanics (Theorems with proof)

1 Wavefunctions and the Schr¨odingerequation 1.1 Particle state and probability 1.2 Operators 1.3 Time evolution of wavefunctions Proposition. The probability density

P (x, t) = |Ψ(x, t)|2

obeys a conservation equation ∂P ∂j = − , ∂t ∂x where i dΨ dΨ∗ j(x, t) = − ~ Ψ∗ − Ψ 2m dx dx is the probability current. Proof. This is straightforward from the Schr¨odinger equation and its complex conjugate. We have ∂P ∂Ψ ∂Ψ∗ = Ψ∗ + Ψ ∂t ∂t ∂t i i = Ψ∗ ~ Ψ00 − ~ Ψ00∗Ψ 2m 2m where the two V terms cancel each other out, assuming V is real ∂j = − . ∂x

4 2 Some examples in one dimensionIB Quantum Mechanics (Theorems with proof)

2 Some examples in one dimension 2.1 Introduction 2.2 Inﬁnite well — particle in a box 2.3 Parity 2.4 Potential well 2.5 The harmonic oscillator

5 3 Expectation and uncertaintyIB Quantum Mechanics (Theorems with proof)

3 Expectation and uncertainty 3.1 Inner products and expectation values Proposition. The operators xˆ, pˆ and H are all Hermitian (for real potentials). Proof. We do xˆ first: we want to show (φ, xψˆ ) = (xφ,ˆ ψ). This statement is equivalent to Z ∞ Z ∞ φ(x)∗xψ(x) dx = (xφ(x))∗ψ(x) dx. −∞ −∞ Since position is real, this is true. To show that pˆ is Hermitian, we want to show (φ, pψˆ ) = (pφ,ˆ ψ). This is equivalent to saying Z ∞ Z ∞ ∗ 0 0 ∗ φ (−i~ψ ) dx = (i~φ ) ψ dx. −∞ −∞ This works by integrating by parts: the difference of the two terms is ∗ ∞ −i~[φ ψ]−∞ = 0 since φ, ψ are normalizable. To show that H is Hermitian, we want to show (φ, Hψ) = (Hφ, ψ), where h2 d2 H = − + V (x). 2m dx2 To show this, it suffices to consider the kinetic and potential terms separately. For the kinetic energy, we just need to show that (φ, ψ00) = (φ00, ψ). This is true since we can integrate by parts twice to obtain (φ, ψ00) = −(φ0, ψ0) = (φ00, ψ). For the potential term, we have (φ, V (ˆx)ψ) = (φ, V (x)ψ) = (V (x)φ, ψ) = (V (ˆx)φ, ψ). So H is Hermitian, as claimed. Proposition (Cauchy-Schwarz inequality). If ψ and φ are any normalizable states, then kψkkφk ≥ |(ψ, φ)|. Proof. Consider kψ + λφk2 = (ψ + λφ, ψ + λφ) = (ψ, ψ) + λ(ψ, φ) + λ∗(φ, ψ) + |λ|2(φ, φ) ≥ 0. This is true for any complex λ. Set (φ, ψ) λ = − kφk2 which is always well-defined since φ is normalizable, and then the above equation becomes |(ψ, φ)|2 kψk2 − ≥ 0. kφk2 So done.

6 3 Expectation and uncertaintyIB Quantum Mechanics (Theorems with proof)

3.2 Ehrenfest’s theorem Theorem (Ehrenfest’s theorem). d 1 hxî = hpî dt Ψ m Ψ d hpî = −hV 0(ˆx)i . dt Ψ Ψ Proof. We have d hxî = (Ψ˙ , xˆΨ) + (Ψ, xˆΨ)˙ dt Ψ 1 1 = HΨ, xˆΨ + Ψ, xˆ H Ψ i~ i~ Since H is Hermitian, we can move it around and get 1 1 = − (Ψ,H(ˆxΨ)) + (Ψ, xˆ(HΨ)) i~ i~ 1 = (Ψ, (ˆxH − Hxˆ)Ψ). i~ But we know 2 2 i (ˆxH − Hxˆ)Ψ = − ~ (xΨ00 − (xΨ)00) + (xV Ψ − V xΨ) = −~ Ψ0 = ~pˆΨ. 2m m m So done. The second part is similar. We have d hpî = (Ψ˙ , pˆΨ) + (Ψ, pˆΨ)˙ dt Ψ 1 1 = HΨ, pˆΨ + Ψ, pˆ H Ψ i~ i~ Since H is Hermitian, we can move it around and get 1 1 = − (Ψ,H(ˆpΨ)) + (Ψ, pˆ(HΨ)) i~ i~ 1 = (Ψ, (ˆpH − Hpˆ)Ψ). i~ Again, we can compute − 2 (ˆpH − Hpˆ)Ψ = −i ~ ((Ψ00)0 − (Ψ0)00) − i ((V (x)Ψ)0 − V (x)Ψ0) ~ 2m ~ 0 = −i~V (x)Ψ. So done.

3.3 Heisenberg’s uncertainty principle Theorem (Heisenberg’s uncertainty principle). If ψ is any normalized state (at any ﬁxed time), then (∆x) (∆p) ≥ ~. ψ ψ 2

7 3 Expectation and uncertaintyIB Quantum Mechanics (Theorems with proof)

Proof of uncertainty principle. Choose α = hxîψ and β = hpîψ, and define

X =x ˆ − α, P =p ˆ − β.

Then we have

2 2 2 (∆x)ψ = (ψ, X ψ) = (Xψ, Xψ) = kXψk 2 2 2 (∆p)ψ = (ψ, P ψ) = (P ψ, P ψ) = kP ψk

Then we have

(∆x)ψ(∆p)ψ = kXψkkP ψk ≥ |(Xψ, P ψ)| ≥ | im(Xψ, P ψ)|

1 h i ≥ (Xψ, P ψ) − (P ψ, Xψ) 2i

1 h i = (ψ, XP ψ) − (ψ, P Xψ) 2i

1 = (ψ, [X,P ]ψ) 2i

~ = (ψ, ψ) 2 = ~. 2 So done.

8 4 More results in one dimensionsIB Quantum Mechanics (Theorems with proof)

4 More results in one dimensions 4.1 Gaussian wavepackets 4.2 Scattering 4.3 Potential step 4.4 Potential barrier 4.5 General features of stationary states

9 5 Axioms for quantum mechanicsIB Quantum Mechanics (Theorems with proof)

5 Axioms for quantum mechanics 5.1 States and observables 5.2 Measurements Proposition. Let Q be Hermitian (an observable), i.e. Q† = Q. Then (i) Eigenvalues of Q are real. (ii) Eigenstates of Q with different eigenvalues are orthogonal (with respect to the complex inner product). (iii) Any state can be written as a (possibly infinite) linear combination of eigenstates of Q, i.e. eigenstates of Q provide a basis for V . Alternatively, the set of eigenstates is complete. Proof. (i) Since Q is Hermitian, we have, by definition,

(χ, Qχ) = (Qχ, χ).

Let χ be an eigenvector with eigenvalue λ, i.e. Qχ = λχ. Then we have

(χ, λχ) = (λχ, χ).

So we get λ(χ, χ) = λ∗(χ, χ). Since (χ, χ) 6= 0, we must have λ = λ∗. So λ is real. (ii) Let Qχ = λχ and Qφ = µφ. Then we have

(φ, Qχ) = (Qφ, χ).

So we have (φ, λχ) = (µφ, χ). In other words, λ(φ, χ) = µ∗(φ, χ) = µ(φ, χ). Since λ 6= µ by assumption, we know that (φ, χ) = 0. (iii) We will not attempt to justify this, or discuss issues of convergence. Proposition. (i) The expectation value of Q in state ψ is X hQiψ = (ψ, Qψ) = λnPn,

with notation as in the previous part.

(ii) The uncertainty (δQ)ψ is given by

2 2 2 2 X 2 (∆Q)ψ = h(Q − hQiψ) iψ = hQ iψ − hQiψ = (λn − hQiψ) Pn. n

10 5 Axioms for quantum mechanicsIB Quantum Mechanics (Theorems with proof)

Proof. P (i) Let ψ = αnχn. Then X Qψ = αnλnχn.

So we have

X X ∗ X (ψ, Qψ) = (αm, χn, αnλnχn) = αnαnλn = λnPn. n,m n

(ii) This is a direct check using the ﬁrst expression.

5.3 Evolution in time Theorem (Ehrenfest’s theorem). If Q is any operator with no explicit time dependence, then d i hQi = h[Q, H]i , ~dt Ψ Ψ where [Q, H] = QH − HQ is the commutator. Proof. If Q does not have time dependence, then d i (Ψ,QΨ) = (−i Ψ˙ ,QΨ) + (Ψ, Qi Ψ)˙ ~dt ~ ~ = (−HΨ,QΨ) + (Ψ, QHΨ) = (Ψ, (QH − HQ)Ψ) = (Ψ, [Q, H]Ψ).

5.4 Discrete and continuous spectra 5.5 Degeneracy and simultaneous measurements

11 6 Quantum mechanics in threeIB dimensions Quantum Mechanics (Theorems with proof)

6 Quantum mechanics in three dimensions 6.1 Introduction 6.2 Separable eigenstate solutions 6.3 Angular momentum Proposition.

(i)[ Li,Lj] = i~εijkLk. 2 (ii)[ L ,Li] = 0.

(iii)[ Li, xˆj] = i~εijkxˆk and [Li, pˆj] = i~εijkpˆk Proof. (i) We ﬁrst do a direct approach, and look at speciﬁc indices instead of general i and j. We have 2 ∂ ∂ ∂ ∂ L1L2 = (−i~) x2 − x3 x3 − x1 ∂x3 ∂x2 ∂x1 ∂x3 2 2 2 2 ∂ ∂ ∂ 2 ∂ ∂ = −~ x2 x3 − x1x2 2 − x3 + x3x1 ∂x3 ∂x1 ∂x3 ∂x2∂x1 ∂x2∂x3 Now note that ∂ ∂ ∂2 ∂ x2 x3 = x2x3 + x2 . ∂x3 ∂x1 ∂x3x1 ∂x1

Similarly, we can compute L2L1 and ﬁnd 2 ∂ ∂ [L1,L2] = −~ x2 − x1 = i~L3, ∂x1 ∂x2 where all the double derivatives cancel. So done. Alternatively, we can do this in an abstract way. We have

LiLj = εiarxˆapˆrεjbsxˆbpˆs

= εiarεjbs(ˆxapˆrxˆbpˆs)

= εiarεjbs(ˆxa(ˆxbpˆr − [ˆpr, xˆb])ˆps) = εiarεjbs(ˆxaxˆbpˆrpˆs − i~δbrxˆapˆs) Similarly, we have

LjLi = εiarεjbs(ˆxbxˆapˆspˆr − i~δasxˆbpˆr) Then the commutator is

LiLj − LjLi = −i~εiarεjbs(δbrxâpˆs − δasxˆbpˆr) = −i~(εiabεjbsxâpˆs − εiarεjbaxˆbpˆr) = −i~((δisδja − δijδas)ˆxapˆs − (δibδrj − δijδrb)ˆxbpˆr) = i~(ˆxipˆj − xˆjpî) = i~εijkLk. So done.

12 6 Quantum mechanics in threeIB dimensions Quantum Mechanics (Theorems with proof)

(ii) This follows from (i) using the Leibnitz property:

[A, BC] = [A, B]C + B[A, C].

This property can be proved by directly expanding both sides, and the proof is uninteresting. Using this, we get

2 [Li, L ] = [Li,LjLj]

= [Li,Lj]Lj + Lj[Li,Lj] = i~εijk(LkLj + LjLk) = 0

where we get 0 since we are contracting the antisymmetric tensor εijk with the symmetric tensor LkLj + LjLk. (iii) We will use the Leibnitz property again, but this time we have it the other way round: [AB, C] = [A, C]B + A[B,C]. This follows immediately from the previous version since [A, B] = −[B,A]. Then we can compute

[Li, xˆj] = εiab[ˆxapˆb, xˆj]

= εiab([ˆxa, xˆj]ˆpb +x â[ˆpb, xˆj]) = εiabxâ(−i~δbj) = i~εijaxâ as claimed. We also have

[Li, pˆj] = εiab[ˆxapˆb, pˆj]

= εiab([ˆxa, pˆj]ˆpb +x ˆa[ˆpb, pˆj]) = εiab(i~δajpˆb) = i~εijbpˆb.

6.4 Joint eigenstates for a spherically symmetric potential

13 7 The hydrogen atom IB Quantum Mechanics (Theorems with proof)

7 The hydrogen atom 7.1 Introduction 7.2 General solution 7.3 Comments