Lecture 04 - Projectors, bras and kets (Schuller's Lectures on Quantum Theory)

Acknowledgments This set of lecture notes accompanies Frederic Schuller’s course on Quantum Theory, taught in the summer of 2015 at the Friedrich-Alexander-Universität Erlangen-Nürnberg as part of the Elite Graduate Programme. The entire course is hosted on YouTube at the following address: www.youtube.com/playlist?list=PLPH7f_7ZlzxQVx5jRjbfRGEzWY_upS5K6 These lecture notes are not endorsed by Dr. Schuller or the University. While I have tried to correct typos and errors made during the lectures (some helpfully pointed out by YouTube commenters), I have also taken the liberty to add and/or modify some of the material at various points in the notes. Any errors that may result from this are, of course, mine. If you have any comments regarding these notes, feel free to get in touch. Visit my blog for the most up to date version of these notes http://mathswithphysics.blogspot.com My gratitude goes to Dr. Schuller for lecturing this course and for making it available on YouTube.

Simon Rea

4

Projectors, bras and kets

4.1

Projectors

Projectors play a key role in quantum theory, as you can see from Axioms 2 and 5. Definition. Let H be a separable Hilbert space. Fix a unit vector e ∈ H (that is, kek = 1) and let ψ ∈ H. The projection of ψ to e is ψ := he|ψie while the orthogonal complement of ψ is ψ⊥ := ψ − ψ . We can extend these definitions to a countable orthonormal subset {ei }i∈N ⊂ H, i.e. a subset of H whose elements are pairwise orthogonal and have unit norm. Note that {ei }i∈N need not be a basis of H. Proposition 4.1. Let ψ ∈ H and let {ei }i∈N ⊂ H be an orthonormal subset. Then (a) we can write ψ = ψ + ψ⊥ , where ∞ X hei |ψiei , ψ :=

ψ⊥ := ψ − ψ

i=0

and we have ∀ i ∈ N : hψ⊥ |ei i = 0. (b) Pythagoras’ theorem holds: kψk2 = kψ k2 + kψ⊥ k2 . Note that this is an extension to the finite-dimensional case. (c) for any γ ∈ span{ei | i ∈ N}, we have the estimate kψ − γk ≥ kψ⊥ k with equality if, and only if, γ = ψ . Proof. First consider the case of a finite orthonormal subset {e0 , . . . , en } ⊂ H. (a) Let ψ and ψ⊥ be defined as in the proposition. Then ψ + ψ⊥ = ψ and   n X hψ⊥ |ei i = ψ − hej |ψiej ei j=0

= hψ|ei i −

n X

hej |ψihej |ei i

j=0 n X = hψ|ei i − hψ|ej iδji j=0

= hψ|ei i − hψ|ei i =0 for all 0 ≤ i ≤ n.

–1–

(b) From part (a), we have  hψ⊥ |ψ i =

n  X n X ψ⊥ hei |ψiei = hei |ψihψ⊥ |ei i = 0. i=0

i=0

Hence, by (the finite-dimensional) Pythagoras’ theorem kψk2 = kψ + ψ⊥ k2 = kψ k2 + kψ⊥ k2 . (c) Let γ ∈ span{ei | 0 ≤ i ≤ n}. Then γ =

Pn

i=0 γi ei

for some γ0 , . . . , γn ∈ C. Hence

kψ − γk2 = kψ⊥ + ψ − γk2

2 n n X X

= ψ⊥ + hei |ψiei − γi e i

i=0

i=0

2 n X

= (hei |ψi − γi )ei

ψ⊥ +

i=0

= kψ⊥ k2 +

n X

|hei |ψi − γi |2

i=0

and thus kψ − γk ≥ kψ⊥ k since |hei |ψi − γi |2 > 0 for all 0 ≤ i ≤ n. Moreover, we have equality if, and only if, |hei |ψi − γi | = 0 for all 0 ≤ i ≤ n, that is γ = ψ . To extend this to a countably infinite orthonormal set {ei }i∈N , note that by part (b) and Bessel’s inequality, we have

X

2 X n

n

hei |ψiei = |hei |ψi|2 ≤ kψk2 .

i=0

i=0

Pn 2 Since |hei ≥ 0, the sequence of partial sums i=0 |hei |ψi| n∈N is monotonically increasing and bounded from above by kψk. Hence, it converges and this implies that |ψi|2

ψ :=

∞ X

hei |ψiei

i=0

exists as an element of H. The extension to the countably infinite case then follows by continuity of the inner product. 4.2

Closed linear subspaces

We will often be interested in looking at linear subspaces of a Hilbert space H, i.e. subsets M ⊆ H such that ∀ ψ, ϕ ∈ M : ∀ z ∈ C : zψ + ϕ ∈ M. Note that while every linear subspace M ⊂ H inherits the inner product on H to become an inner product space, it may fail to be complete with respect to this inner product. In other words, non every linear subspace of a Hilbert space is necessarily a sub-Hilbert space. The following definitions are with respect to the norm topology on a normed space and can, of course, be given more generally on an arbitrary topological space.

–2–

Definition. Let H be a normed space. A subset M ⊂ H is said to be open if ∀ ψ ∈ M : ∃ r > 0 : ∀ ϕ ∈ H : kψ − ϕk < ε ⇒ ϕ ∈ M. Equivalently, by defining the open ball of radius r > 0 and centre ψ ∈ H Br (ψ) := {ϕ ∈ H | kψ − ϕk < r}, we can define M ⊂ H to be open if ∀ ψ ∈ M : ∃ r > 0 : Br (ψ) ⊆ M. Definition. A subset M ⊂ H is said to be closed if its complement H \ M is open. Proposition 4.2. A closed subset M of a complete normed space H is complete. Proof. Let {ψn }n∈N be a Cauchy sequence in the closed subset M. Then, {ψn }n∈N is also a Cauchy sequence in H, and hence it converges to some ψ ∈ H since H is complete. We want to show that, in fact, ψ ∈ M. Suppose, for the sake of contradiction, that ψ ∈ / M, i.e. ψ ∈ H \ M. Since M is closed, H \ M is open. Hence, there exists r > 0 such that ∀ ϕ ∈ H : kϕ − ψk < r ⇒ ϕ ∈ H \ M. However, since ψ is the limit of {ψn }n∈N , there exists N ∈ N such that ∀ n ≥ N : kψn − ψk < r. Hence, for all n ≥ N , we have ψn ∈ H \ M, i.e. ψn ∈ / M, contradicting the fact that {ψn }n∈N is a sequence in M. Thus, we must have ψ ∈ M. Corollary 4.3. A closed linear subspace M of a Hilbert space H is a sub-Hilbert space with the inner product on H. Moreover, if H is separable, then so is M. Knowing that a linear subspace of a Hilbert space is, in fact, a sub-Hilbert space can be very useful. For instance, we know that there exists an orthonormal basis for the linear subspace. Note that the converse to the corollary does not hold: a sub-Hilbert space need not be a closed linear subspace. 4.3

Orthogonal projections

Definition. Let M ⊆ H be a (not necessarily closed) linear subspace of H. The set M⊥ := {ψ ∈ H | ∀ ϕ ∈ M : hϕ|ψi = 0} is called the orthogonal complement of M in H. Proposition 4.4. Let M ⊆ H be a linear subspace of H. Then, M⊥ is a closed linear subspace of H.

–3–

Proof. Let ψ1 , ψ2 ∈ M⊥ and z ∈ C. Then, for all ϕ ∈ M hϕ|zψ1 + ψ2 i = zhϕ|ψ1 i + hϕ|ψ2 i = 0 and hence zψ1 + ψ2 ∈ M. Thus, M⊥ is a linear subspace of H. It remains to be shown that it is also closed. Define the maps fϕ : H → C ψ 7→ hϕ|ψi. Then, we can write \

M⊥ =

preimfϕ ({0}).

ϕ∈M

Since the inner product is continuous (in each slot), the maps fϕ are continuous. Hence, the pre-images of closed sets are closed. As the singleton {0} is closed in the standard topology on C, the sets preimfϕ ({0}) are closed for all ϕ ∈ M. Thus, M⊥ is closed since arbitrary intersections of closed sets are closed. Note that by Pythagoras’ theorem, we have the decomposition H = M ⊕ M⊥ := {ψ + ϕ | ψ ∈ M, ϕ ∈ M⊥ } for any closed linear subspace M. Definition. Let M be a closed linear subspace of a separable Hilbert space H and fix some orthonormal basis of M. The map PM : H → M ψ 7→ ψ is called the orthogonal projector to M. Proposition 4.5. Let PM : H → M be an orthogonal projector to M ⊆ H. Then 2 =P (i) PM ◦ PM = PM , sometimes also written as PM M

(ii) ∀ ψ, ϕ ∈ H : hPM ψ|ϕi = hψ|PM ϕi (iii) PM⊥ ψ = ψ⊥ (iv) PM ∈ L(H, M). Proof. Let {ei }i∈I and {ei }i∈J be bases of M and M⊥ respectively, where I, J are either finite or countably infinite, such that {ei }i∈I∪J is a basis of H (in the latter case, we think of I as having a definite ordering).

–4–

(i) Let ψ ∈ H. Then X  PM (PM ψ) := PM hei |ψiei i∈I  X X ;= ej hei |ψiei ej j∈I i∈I XX = hei |ψihej |ei iej j∈I i∈I

X = hei |ψiei i∈I

=: PM ψ. (ii) Let ψ, ϕ ∈ H. Then  X hPM ψ|ϕi := hei |ψiei ϕ i∈I

=

X

hei |ψihei |ϕi

i∈I

X = hei |ϕihψ|ei i i∈I   X = ψ hei |ϕiei i∈I

=: hψ|PM ϕi. (iii) Let ψ ∈ H. Then PM ψ + PM⊥ ψ =

X

hei |ψiei +

i∈I

X X hei |ψiei = hei |ψiei = ψ. i∈J

i∈I∪J

Hence PM⊥ ψ = ψ − PM ψ = ψ − ψ =: ψ⊥ . (iv) Let ψ ∈ H. Then, by Pythagoras’ theorem, kPM ψk kψ k kψk − kψ⊥ k = sup = sup ≤1