## Quantum Mechanics for Dummies

Intro
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Quantum mechanics (QM) describes sets of particles/waves as point state-vectors in a multidimensional space where each co-ordinate is a complex number (refine). QM does not deal with relativistic physics, in order to do so, it was extended to deal with fields rather than particles. Two Quantum Fields Theories exist describing three of the four fundamental forces (electromagnetism, weak nuclear, strong nuclear, gravity). [There is no quantum field theory of gravity, yet]:

1.The electroweak theory which combines Quantum Electrodynamics (QED) describing electromagnetism, and a quantum field theory of the weak nuclear force.

2. Quantum Chromodynamics (QCD) which describes the strong nuclear force.

QM is pointless to try to grasp without the math, since the quantum realm (before collapse into the classical world we perceive) is something that is defined by abstract mathematics that defy experience. Unfortunately the math is intimidating and requires simple explanations which you won’t get on Wikipedia.

This may seem like a futile exercise for someone with no science education beyond high school, but here is how I tried - If I can you can. (Much of this is based on lectures by Leonard Susskind or Wojciech Zurek).

Most of the problems of interpretation of Quantum theories concern the clash of the determinism of the Schrodinger wave equation with the quantum randomness of the collapse into the classical realm.

Below are the axioms which quantum theory as a whole rests on. Some of the concepts introduced are explained after, in the notes on Quantum Mechanics.

Quantum theory based on 2 axioms which establish the formal structure, but do not say anything about collapse or probabilities:
(i) State of a quantum system is represented by a vector in its Hilbert space HsubS implies =>quantum superposition.
(ii) Evolutions are unitary (i.e., generated by Schrodinger equation) implies => unitarity of quantum evolutions

A third and fourth axiom tie these abstract state vectors in Hilbert space to experimental data:
(iii) Immediate repetition of a measurement yields the same outcome (this is idealized and difficult in practice).
(iv) Measurement outcome is one of the orthonormal states – eigenstates of the measured observable (the collapse postulate, where controversy lies).

A fifth axiom is generally referred to as Born’s rule:

(v) Probability pk of ﬁnding an outcome |sk ⟩ in a measurement of a quantum system that was previously prepared in the state |ψ⟩ is given by |⟨s|ψ⟩|^2.

A further axiom, axiom (o) ‘The Universe consists of systems’ is often omitted from textbooks as obvious. However, it is strictly necessary since in absence of systems (a universe divisible into parts) the measurement problem disappears.

Zurek:

Starting from a general state |ψS ⟩ in a Hilbert space of the system (axiom (i)), an initial state |A0 ⟩ of the apparatus, and assuming unitary evolution (axiom (ii)) one is led to a superposition of outcomes, which contradicts axiom (iv)

In other words, the math predicts that the universe contains all possible outcomes but experimental measurement only ever yields one. This requires an ‘interpretation, and the standard one is the Copenhagen Interpretation:

Which says that the measurer is classical and that the collapse occurs on the boundary between the quantum superimposed axiom (i) and (ii) and the classical.

Without worrying about the metaphysical issues of interpretation at the quantum/classical boundary, there is plenty of useful stuff that can be done with the math of QM. This is based on linear algebra, one of the concepts of linear algebra is that of a vector space, where in QM the vectors represent the state of a system. States aren’t measurable, but features like momentum are, and these measurables are represented by hermitian linear operators.

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History

In classical physics the intensity (energy) of a light beam of given frequency can be arbitratily weak (frequency and wavelength are independent of energy). But in quantum mechanics it cannot be less than 1 photon. This implies the uncertainty principal since there is a limit to how ‘gently’ you can measure a system.

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Vector Spaces

In classical physics a collection of states form a set of points in phase space. The core logic of classical physics in phase space is set theory, a set of points where a point is a state, and where Boolean operations on the set dictate the logic e.g. intersection AND and union OR.

In QM states do not form sets, they are not points but rather vectors in a vector space (a vector space over complex numbers or a Hilbert space.). Thus QM is not based on Boolean logic and the concept of AND and OR are quite different.

Imaginary numbers are well known, but to refresh: Imaginary numbers are the square root of a negative number, Complex numbers are imaginary numbers written as two components (a real and imaginary part) x +iy where i is the square root of -1. (NB. because these numbers can be plotted as a two dimensional plane and because complex numbers are themselves a vector space (you can multiply a complex number by a complex number to get new and you can add two together to get new - they are the one dimensional vector over the complex number space), don’t get confused later by the complex plane and multi-dimensional complex number spaces (i.e. between the dimensionality n and the numbers)).

The ordinary use of the term vector as a pointer, an arrow with a length is merely a specific use of the much more general and abstract term vector that we use here, called ket vectors where the equiv of a point ‘a’ is denoted by ‘|a>’.

Hilbert spaces originally described the vector space over the complex numbers that represent a collection of complex number functions of real numbers (i.e ψ(χ) where ψ is complex and x is real). These can be multi-dimensional. So Hilbert spaces are n dimensional complex vector spaces.

(The Dimension of a vector space is the minimum number of vectors needed so that you can write any vector as a sum with coefficients of that minimum number. In column vectors number of items gives the dimension.
We can think of functions as vectors in a complex vector space, i.e ψ(x) ⇒ |ψ> (⇒ symbol means ‘think of them as’). You need an infinite number of functions to write all functions, therefore functions are an infinite dimensional vector space.)

The logic of vectors in Hilbert space:

(i) Given a vector you can multiply it by a complex number and you get a new vector: α|a> = |b>
(This is very different from classical mechanics where it simply does not make sense to multiply points in a set by a number. e.g. heads and tails - it doesn’t mean anything to multiply heads by three).
(ii) You can add two vectors to get a new vector: |a> + |b> = |c>

(Real functions are a vector space over the real numbers, but you cannot multiply a real function by a complex number and get back a real function (you get back a complex function), so real functions are not a vector space over the complex numbers).

The dimension n is non mixable since there are no obvious rules for adding vectors of different dimensions, so they do not form a vector space.

For every vector space there is a dual vector space or conjugate (i.e. |A> ⇔ <a| (⇔symbol means ‘for every’)), since for every complex number there is a complex conjugate which is a mapping of the number so that it is reflected in the x axis
i.e. z=x +iy -> x - iy, called z*. The dual is represented by a bra vector: <A|. i.e. |A>* = <A|, or ψ*(x) ⇔ ψ(x) when represented as complex conjugates.

Notation: column (ket) vectors are represented by rows of (bra) vectors.
Conjugates of sums are straightforward: |A> + |B> ⇔ <A| + <B|, but Remember to conjugate a complex coefficient (i.e. both parts) in multiplication, i.e. α|A> ⇔ <a |α* NOT α|A> ⇔ α<A|.

Inner product of two vector spaces:
This is the analog of ordinary vector dot products (the dot product is the projection of one vector onto the others axis and then the two multiplied. i.e. ab cos angle between them), it is represented by ‘<a|b>’ and has the following abstract properties.

Properties:

1. The inner product of <A| with a product: <a|(β|B>) = β<a|B> (i.e if you double the length of |B> and not <a|, then you double the length of the inner product).
2. The inner product of <A| with a sum: <A| [ |B> + |C> ] = <A|B> + <A|C>
3. <A|B> is related to but not identical to <B|A>* (the conjugate), <A|A> = <A|A>* = a number not a vector.
(i.e. Numbers which are the complex conjugates of themselves are real numbers (they are their own reflections in the x axis of the complex plane, therefore they lie on the x axis with no complex component)).
4. <A|A>* = z*z = (x +iy)(x -iy) = x^2 + y^2: is always positive.

Rules:

1. The inner product of continuous functions, <φ|ψ> = ∫dx φ*(x)ψ(x)
2. The inner product of discrete vector arrays for a column vectors matrix style product with row of conjugates i.e. :
a (b*, b*, b*)
a
a

is just the sum of the product of the components b*a + b*a + b*a.

Basis vectors (useful because you can easily expand any vector in terms of a basis vector).
Basis vectors are unit vectors orthogonal to each other to represent other vectors in their terms. (In everyday vectors they are like two 1 unit lines at right angles (with any rotation, normally horizontal and vertical)).

Take a set of vectors |bsubi> where i is the dimension and bsubi is a collection of i vectors.
The basis vector properties:
1. They are normalized i.e. they are of unit length and the square of their vector length (dot product) is 1 i.e. <bsubi|bsubi> = 1
2. They are orthogonal (i.e. the projection of one onto the other is zero like x and y axes in everyday vectors)
<bsubi |bsubj> = 0 they are orthogonal.
If both normalized and orthogonal, then called orthonormal.

1 and 2 : = 0 (if j not= i) and = 1 (if j = i)
can be written:
<bsubi|bsubj> = δsubisubjm

It should be possible to take any vector V and express it as the sum over the basis vectors i of coefficients (Vsubi) times the i basis vector. i.e. |V> = ∑ Vsubi |bsubi> for all i. (i.e. V1b1 +V2B2 …)
Skipping a couple of steps:
<bsubi|V> = Vsubi.

In general: |V> = ∑ |subi><bsubi|V>
(The above crops up a lot).

What is the connection between states in QM and vector spaces?

Take a classical phase space of 2 states, say heads and tails. You could consider each to be described by an orthogonal vector, |H> |T>, but it wouldn’t be useful, classical phase spaces are based upon sets and boolean algebra, the notion of things like adding two points together (heads and tails) as vectors to get a third point ( a headytail) wouldn’t mean anything. In quantum theory, a combination (superimposition) of states is meaningful, so they can be combined as vectors.

Postulate 1: In QM, if you can differentiate between two superimposed states, say αsubh|H> + αsubt|T> with a single observation, then the vector components are orthogonal i.e. <H|T> = 0.

(If we make |H> and |T> unit length, i.e. <H|H> = <T|T> = 0 we can form a basis vector).

The mixed state gives the probability of finding one or the other state in observation, given by the coefficients (called the probability amplitude), αsubh or αsubt. Since the probability is a positive real number whereas the probability amplitudes can be positive, negative or complex, the Probability of H, Psubh = αsubh* αsubh
Because certainty = 1 in probability, αsubh* αsubh + αsubt* αsubt = 1 (which is the same as the probability of the inner product of the headytail vector with itself ).
Postulate 2: Any properly defined state of a system should have unit length, i.e. be normalized since probabilities should add up to 1.

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Operators
Operations that you do on vectors. Specifically linear operators, like reflect in the x axis, scale by a number etc. not non-linear operators like square the length etc. (Notation: Bold number or number with a caret on top.)

Linear operators correspond to quantities that can be measured, ‘observables’ rather than states. You can measure the headsness or tailsness of a coin, so the headsness or tailsness is represented by a linear operator.

Properties:
1. A linear operator that acts on a vector gives another vector. i.e. L |A> -> |B>
and L α|A> = α L |B>
2. L { |A> |B> } = L|A> + L|B>

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