PHYS20602 Vector Spaces for Quantum Mechanics

Aims:
To introduce the idea of abstract vector spaces and to use them as a framework to solve problems in quantum mechanics.

Learning Outcomes:
On completion successful students will be able to:


 * 1) define a linear vector space and its inner product.
 * 2) solve matrix eigenvalue problems.
 * 3) use Dirac notation to represent quantum-mechanical states and their properties.
 * 4) use 2-component complex vectors to describe spin-½ systems.
 * 5) construct eigenstates of total angular momentum operators
 * 6) solve the simple harmonics oscillator using creation and annihilation operators.
 * 7) apply these techniques to problems from atomic, nuclear or particle physics.

Sets
A set is a collection of elements, and is defined with the notation:

A={1, 2, 3}

Where A is the label of the set, and (in this case) 1, 2, and 3 are the elements in the set. These could just have easily been letters, colours, functions or even other sets.

In this example, it can be said that 1∈A, ie 1 belongs to the set A.

Alternatively, 4∉A. That is, 4 is not a member of the set A.


 * Important Sets
 * ℕ - All natural numbers: {1, 2, 3, . . .}.
 * ℤ - Both positive and negative integers: {..., −2, −1, 0, 1, 2, ...}.
 * ℝ - All real numbers (all rational and irrational numbers. Including 0 and negatives).
 * ℂ - All complex numbers: {x + iy | x,y∈ℝ}.
 * ∅ - The empty set: {}.


 * Combinations

Using A defined above, and two other sets, B={3, 4, 5} and C={6, 7}; we can define the following:


 * Union - A ∪ B = {1, 2, 3, 4, 5}
 * Take the elements of both sets (without duplication).


 * Intersections - A ∩ B = {3} ; A ∩ C = ∅
 * Take the elements common to both sets.


 * Subset - {2, 3} ⊂ A
 * A subset contains elements from the other set, but no others.


 * Superset - A ⊃ {2, 3}
 * A superset contains all the elements of the other set, but others besides.
 * Where one set is a subset of another, the reverse can be said that the other set is a superset of it.

Groups
A group is a system, notated [G,•], where G is a set, and • is an operation (for example, +).

To be a group, the following rules apply:
 * 1) The set is closed under •. That is, a • b∈G, for any a, b∈G. In other words, if the operator is applied to two members of the set, the resultant must also be a member.
 * 2) The operation is associative. That is, a • (b • c) = (a • b) • c for any a, b, c∈G.
 * 3) There is an identity element e∈G, such that a • e = e • a = a, for all a∈G.
 * 4) Every a∈G has an inverse element a$$^{-1}$$, such that a • a$$^{-1}$$ = a$$^{-1}$$ • a = e.

If the operation is commutative, where a • b = b • a, then the group is abelian.

NB. In the notation above, • and $$^{-1}$$ are placeholders representing the operation and the inverse. These take different forms for each Group.

For example, in the group [ℝ,+], the operator is addition, +, the identity element e = 0, and the inverse of an element a$$^{-1}$$ is $$-$$a. This group is also abelian.

Vector Spaces
A complex vector space is a set, written V(ℂ), of elements called vectors, where
 * 1) There exists an operation (+) such that [V(ℂ), +] is an arbelian group. The identity element is written as 0 and is known as the zero vector. The inverse of vector $$x$$ is written as $$-x$$. (Note that in this case, $$-x$$ is just a name, and not an operation. However for convenience an minus operation is defined such that a $$-$$ b = a + ($$-$$b ) )
 * 2) For any complex numbers α, β∈ℂ and vectors $$x$$, $$y$$∈V(ℂ), products of any complex number with vectors are vectors in V(ℂ). ie α$$x$$∈V(ℂ). Also
 * 3) $$\alpha ( \beta x ) = ( \alpha$$ x $$\beta ) x$$
 * 4) $$1x = x $$
 * 5) $$\alpha (x + y) = \alpha x + \alpha y$$
 * 6) $$(\alpha + \beta ) x = \alpha x + \beta y$$

For a real vector space V(ℝ), the same applies, except α, β∈ℝ, and $$x$$, $$y$$∈V(ℝ).

Dirac Notation
In this course, we use Dirac notation to represent vectors so as to stop the confusion between abstract vectors and simple numbers.

Dirac Notation consists of two parts, the Bra and the Ket. Both are formed from vertical lines and angled brackets.

In this system instead of writing a vector as symbols such as $$\psi \;$$, we write $$|\psi\rangle$$, which is the ket.

As a comparison, if we were to write this in terms of matrix notation, we would have: $$\left(\begin{array}{c} \psi_{1}\\ \vdots\\ \psi_{N} \end{array}\right)=[\psi] $$

The bra is the dual of $$\psi \,$$, written as $$\psi^{*} \,$$.

Again as a comparison, if we were to write this in matrix notation we would have: $$\left(\begin{array}{ccc} \psi_{1}^{*} & \ldots & \psi_{N}^{*}\end{array}\right)=\left([\psi]^{T}\right)^{*}$$.

When writing in this method, it can easily be seen from matrix multiplication, that the inner product $$\langle a|b\rangle$$ will give a single complex number.


 * Comparing a bra and a ket

If we define a ket $$|c\rangle=\alpha|a\rangle$$, then the bra $$\langle c|=\alpha^{*}\langle a|$$.

Linear Independence
A set of vectors
 * $$\left\{ |i\rangle\right\} _{i=1}^{N}$$

is linearly independent if:
 * $${\underset{i}{\sum}}{\displaystyle a_{i}|i\rangle=0}$$

which implies all $${a_{i}=0}$$, otherwise the $$\left\{ |i\rangle\right\}$$ are linearly dependent.

For linearly dependent vectors, some members of the set can be represented in terms of the rest. For example, we have:
 * $$a_{1}|1\rangle+a_{2}|2\rangle+...+a_{j}|j\rangle+...+a_{n}|n\rangle=0$$

Therefore:
 * $$|j\rangle=-\frac{1}{a_{j}}\underset{i\neq j}{\sum}a_{i}|i\rangle$$

(assuming that $$\scriptstyle a_{j}\neq0$$)

Dimensions
A vector space has a dimension N if it can accommodate N number of linearly independent vectors.

It is infinite-dimensional if it can accommodate N linearly independent vectors for all N.

Bases
In vector spaces, a basis is defined as set of linearly independent vectors $$\left\{|x_i\rangle\right\}$$ in a vector space V, where every vector in V is a linear combination of $$|x_i\rangle$$:


 * $$|V\rangle=\overset{M}{\underset{i=1}{\sum}}v_{i}|x_{i}\rangle$$

For example, in real 3D space, we can use $$\scriptstyle\hat{i}$$, $$\scriptstyle\hat{j}$$, and $$\scriptstyle\hat{k}$$ to represent any vector. However, M must be a finite number, as we have no way of telling if an infinite sum converges.

Relevant Theorems: 1.1, 1.2

Coordinates
From the vector space definition, $$|a\rangle=\sum a_{i}|i\rangle$$ and $$|b\rangle=\sum b_{i}|i\rangle$$.


 * Let $$|c\rangle = |a\rangle + |b\rangle$$,


 * it goes that:
 * $$|c\rangle=\sum a_{i}|i\rangle+\sum b_{i}|i\rangle=\sum(a_{i}+b_{i})|i\rangle=\sum c_{i}|i\rangle$$
 * ie. $$c_i = a_i + b_i \,$$ : we just add coordinates together when adding vectors.


 * Similarly,
 * $$k|a\rangle=k\sum a_{i}|i\rangle=\sum ka_{i}|i\rangle$$
 * ie, scalar x vector correstpongs to multiplying all coordinates by the scalar.


 * A set formed by a sequence of N complex numbers, ℂ, is a vector space, and coordinates of an abstract vector in N-Dimensions form such a sequence. Therefore for a given basis, we have a unique one-to-one map between an N-D complex vector space and ℂN.
 * ie, for every $$|v\rangle$$, there is a corresponding sequence & vice versa.


 * Symbolically, $$|v\rangle \rightarrow (v_1, v_2, ... v_N) $$


 * Any operation that is applied on an abstract vector space, is mirrored in the sequence ℂN. Therefore, it is said that ℂN is isomorphic to all N-D vector spaces.

Inner Product
For all real 3D spaces, we can use the dot product to define the length and angle between two vectors. However, this is problematic for complex spaces as it can result in a complex or negative definition. The sollution is to define another operation, the inner product.

The inner product of two abstract vectors $$|a\rangle$$ and $$|b\rangle$$ is written as $$\langle a|b\rangle$$, and is a complex number that obeys the following three axioms: $$
 * 1) $$\langle a|b\rangle = \langle b|a\rangle^{*}$$
 * 2) *skew symmetry
 * 3) $$\langle a|a\rangle \geq0$$
 * 4) *equality if, and only if, $$\scriptstyle |a\rangle = |0\rangle$$ (positive definitiveness)
 * 5) $$\langle a|\left(\alpha|b\rangle+\beta|c\rangle\right)=\alpha\langle a|b\rangle+\beta\langle a|c\rangle
 * 1) *where α, β∈ℂ - Linearity in ket.

NB: axiom 1) requires $$\langle a|a\rangle\in\mathbb{R}$$


 * Example of calculation

Let $$|c\rangle = \alpha|a\rangle+\beta|b\rangle$$ What is $$\langle c|d\rangle$$ ?

Using the axioms in the following steps:


 * 1) $$\langle c|d\rangle = \langle d|c\rangle^{*}$$


 * 3) $$\langle d|c\rangle^{*} = \left(\alpha\langle d|a\rangle+\beta\langle d|b\rangle\right)^{*}$$


 * 1) $$\langle c|d\rangle = \alpha^{*}\langle a|d\rangle+\beta^{*}\langle b|d\rangle$$


 * Notes on Inner Product:
 * Inner product are 'anti-linear' on the left
 * Two products are orthogonal if their inner product is zero (as with the dot product)
 * The norm (or length) of a vector $$|v\rangle$$ is given by:
 * $$||v||=\sqrt{\langle v|v\rangle }$$

Orthonormal Basis
If a set of vectors, all of unit norm (length), are orthogonal to each other, then it is called an orthonormal set. Vectors in an orthonormal set are linearly independent.


 * Proof:

If
 * $$\underset{i}{\sum}a_{i}|i\rangle=|0\rangle$$

Then
 * $$\langle j|0\rangle=\underset{i}{\sum}a_{i}\langle j|i\rangle$$
 * $$=\underset{i}{\sum}a_{i}\delta_{ij}$$
 * $$=a_j=0 \,$$

For all $$j$$, therefore it is only satisfied when all $$a_i=0$$, which is the definition of a linearly independent vector.

If there are enough vectors in the orthonormal set to make a basis, it is called an orthonormal basis or complete orthonormal set.

Useful Theroem: PHYS20602 Vector Spaces for Quantum Mechanics

Using the linearity on the right hand side and anti-linearity on the left hand side, we can show that for Orthonormal Basis:
 * $$\langle a|b\rangle=\underset{i}{\sum}a_{i}^{*}b_{i}$$

and so also:
 * $$\langle a|a\rangle=\underset{i}{\sum}a_{i}^{*}a_{i}=\underset{i}{\sum}|a_{i}|^{2}$$

NB: The intemediatory step in this proof is a mess involving two sums and the innerproduct of two orthogonal basis.

Similarly, $$\langle i|a\rangle = a_i$$

A vector space with an inner product is called an inner product space, and is the only kind of vector space dealt with in this course.

1.1
Every vector in V has a unique set of coefficients (coordinates) in terms of a given basis.


 * Proof

Consider $$|V\rangle=\overset{M}{\underset{i=1}{\sum}}v_{i}|x_{i}\rangle$$. Suppose there was another set of coefficients $$\{w_{i}\} $$. Then:
 * $$0=|V\rangle-|V\rangle=\sum v_{i}|x_{i}\rangle-\sum w_{i}|x_{i}\rangle$$

which can be rearranged to
 * $$0=\sum(v_{i}-w_{i})|x_{i}\rangle$$

But, by definition, $$\scriptstyle |x_{i}\rangle$$ are linearly independent vectors. Thus, for all i:
 * $$v_{i}-w_{i}=0\;$$

and thus all coefficients are unique.

1.2
Every basis in a vector space of dimension N $$\scriptstyle\left(V^{N}\right)$$ has N elements.

Section 1: Maths of Vector Spaces
1.1 Preliminaries

1.1.1 Sets

A set is a collection of elements. For example A = {1, 2, 3} where A is the label of the set and 1, 2, 3 are the elements. Elements are anything; numbers, points vectors, functions sets etc. When writing about sets we may say

"1 ∈ A " meaning "1 is in A"

and "4 ∉ A " meaning "4 is not in A".

 Important sets:

ℕ natural numbers {1, 2, 3, ...}

ℤ integers {0, ±1, ±2, ±3 ...}

<span style="color: rgb(0, 0, 0); font-family: sans-serif; font-style: normal; line-height: 19px; ">ℝreal numbers {anything B> 0, including decimals}

<p style="text-align: left;"><span style="color: rgb(0, 0, 0); font-family: sans-serif; font-style: normal; line-height: 19px; ">ℂcomplex numbers {x + iy | x, y <span style="border-style: initial; border-color: initial; font-style: normal; color: rgb(0, 0, 0); font-family: sans-serif; line-height: 19px; ">∈ <span style="color: rgb(0, 0, 0); font-family: sans-serif; font-style: normal; line-height: 19px; ">ℝ}

<p style="text-align: left;"><span style="color: rgb(34, 34, 34); font-family: arial, sans-serif; font-size: small; font-style: normal; line-height: 16px; ">ϕ <span style="color: rgb(34, 34, 34); font-family: arial, sans-serif; font-size: small; font-style: normal; line-height: 16px; ">The Empty Set { }

<p style="text-align: left;"><span style="color: rgb(34, 34, 34); font-family: arial, sans-serif; font-size: small; font-style: normal; line-height: 16px; "> Combinations: Let B = {3, 4, 5} and C = {6, 7}

<p style="text-align: left;">Union - A <span style="color: rgb(0, 0, 0); font-family: sans-serif; font-style: normal; line-height: 19px; white-space: nowrap; ">∪ B = {1, 2, 3, 4, 5}

<p style="text-align: left;">Intersection - A <span style="color: rgb(0, 0, 0); font-family: sans-serif; font-style: normal; line-height: 19px; ">∩ B = {3}

<p style="text-align: left;">Subsets - {2, 3} <span style="color: rgb(0, 0, 0); font-family: sans-serif; font-style: normal; line-height: 19px; "> <span style="color: rgb(0, 0, 0); font-family: sans-serif; font-style: normal; line-height: 19px; ">⊂ A or A <span style="color: rgb(0, 0, 0); font-family: sans-serif; font-style: normal; line-height: 19px; ">⊃ {2, 3}

<p style="text-align: left;"><span style="color: rgb(0, 0, 0); font-family: sans-serif; font-style: normal; line-height: 19px; ">[N.B. by definition A <span style="color: rgb(0, 0, 0); font-family: sans-serif; font-style: normal; line-height: 19px; ">⊂ A]

<p style="text-align: left;"><span style="color: rgb(0, 0, 0); font-family: sans-serif; font-style: normal; line-height: 19px; "> Groups - See handout 1.

<p style="text-align: left;"> <span style="color: rgb(0, 0, 0); font-family: sans-serif; font-style: normal; line-height: 19px; ">Examples of Groups

<p style="text-align: left;"> The integers under addition [Z, +]

<p style="text-align: left;"> e = 0 and a <span style="color: rgb(34, 34, 34); font-family: arial, sans-serif; font-size: small; font-style: normal; line-height: 16px; ">⁻¹ = a <span style="color: rgb(34, 34, 34); font-family: arial, sans-serif; font-size: small; font-style: normal; line-height: 16px; ">→abelian.

<p style="text-align: left;"> <span style="color: rgb(34, 34, 34); font-family: arial, sans-serif; font-size: small; font-style: normal; line-height: 16px; ">Comments on Vector Spaces <span style="color: rgb(34, 34, 34); font-family: arial, sans-serif; font-size: small; font-style: normal; line-height: 16px; ">

<p style="text-align: left;">Do not assume anything about abstract vectors that is not given in the definitions. The rules apply to many things apart from "arrow vectors". So far we have not discussed the "length" or the "angle" between two vectors.

<p style="text-align: left;">Examples: <li style="text-align: left;">3 - Vectors r = (x, y, z) form a real vector space</li></li>

<li style="text-align: left;">Real numbers form a very simple vector space</li></li>

<li style="text-align: left;">The set <span style="color: rgb(0, 0, 0); font-family: sans-serif; font-style: normal; line-height: 19px; ">ℂ <span style="color: rgb(0, 0, 0); font-family: sans-serif; font-style: normal; line-height: 19px; text-align: -webkit-auto; ">ᴺ of sequences of N complex numbers (c1, c2, c3, ... cN) form a complex vector space, where "+" is an ordinary matrix addition 0 = (0, 0, ... 0) and its inverse <span style="color: rgb(0, 0, 0); font-family: sans-serif; font-style: normal; line-height: 19px; text-align: -webkit-auto; ">(-c <sub style="border-style: initial; border-color: initial; font-style: normal; color: rgb(0, 0, 0); font-family: sans-serif; text-align: -webkit-auto; ">1 <span style="color: rgb(0, 0, 0); font-family: sans-serif; font-style: normal; line-height: 19px; text-align: -webkit-auto; ">, -c <sub style="border-style: initial; border-color: initial; font-style: normal; color: rgb(0, 0, 0); font-family: sans-serif; text-align: -webkit-auto; ">2 <span style="color: rgb(0, 0, 0); font-family: sans-serif; font-style: normal; line-height: 19px; text-align: -webkit-auto; ">, -c <sub style="border-style: initial; border-color: initial; font-style: normal; color: rgb(0, 0, 0); font-family: sans-serif; text-align: -webkit-auto; ">3 <span style="color: rgb(0, 0, 0); font-family: sans-serif; font-style: normal; line-height: 19px; text-align: -webkit-auto; ">, ... -c <sub style="border-style: initial; border-color: initial; font-style: normal; color: rgb(0, 0, 0); font-family: sans-serif; text-align: -webkit-auto; ">N <span style="color: rgb(0, 0, 0); font-family: sans-serif; font-style: normal; line-height: 19px; text-align: -webkit-auto; ">). </li></li>

<li style="text-align: left;"><span style="color: rgb(0, 0, 0); font-family: sans-serif; font-style: normal; line-height: 19px; text-align: -webkit-auto; ">The set P of polynomials f(x) = a0 + a1x + a2x2 +... with ai <span style="color: rgb(0, 0, 0); font-family: sans-serif; font-style: normal; line-height: 19px; ">∈  <span style="color: rgb(0, 0, 0); font-family: sans-serif; font-style: normal; line-height: 19px; ">ℂ and x  <span style="border-style: initial; border-color: initial; font-style: normal; color: rgb(0, 0, 0); font-family: sans-serif; line-height: 19px; ">∈ <span style="color: rgb(0, 0, 0); font-family: sans-serif; font-style: normal; line-height: 19px; ">ℝ, form a complex vector space (0 is the polynomial with all ai = 0) </li></li>

<li style="text-align: left;"><span style="color: rgb(0, 0, 0); font-family: sans-serif; font-style: normal; line-height: 19px; ">Set if 2x2 complex matrices with compenents a, b, c, d <span style="color: rgb(0, 0, 0); font-family: sans-serif; font-style: normal; line-height: 19px; ">∈  <span style="color: rgb(0, 0, 0); font-family: sans-serif; font-style: normal; line-height: 19px; ">ℂ form a complex vector spaceunder matrix addition. </li><p style="text-align: left;"><span style="color: rgb(0, 0, 0); font-family: sans-serif; font-style: normal; line-height: 19px; ">​

<p style="text-align: left;"><span style="color: rgb(0, 0, 0); font-family: sans-serif; font-style: normal; line-height: 19px; ">'''1.2.3. Notation'''

<p style="text-align: left;"><span style="color: rgb(0, 0, 0); font-family: sans-serif; font-style: normal; line-height: 19px; ">In future we will use Dirac notation for these abstract vectors. Instead of writing a vector as x we write it as |x <span style="color: rgb(34, 34, 34); font-family: arial, sans-serif; font-size: small; font-style: normal; line-height: 16px; ">〉 <span style="color: rgb(34, 34, 34); font-family: arial, sans-serif; font-size: small; font-style: normal; line-height: 16px; ">(a 'ket' lablel). </li>