Quantum Mechanics - Foundations

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THIS PAGE IS WORK IN PROGRESS - It is becoming an elemntary text on quantum mechanics! WHICH IS DEFINITELY NOT WHAT WAS INTENDED.

1.0      Mathematical Prelimenaries

This section gives a very informal overview of the some of the mathematics required to understand Quantum Mechanics. If you do not feel comfortable with any of the topics on this page, it probably makes sense to go swat up on those topics elsewhere and come back at a later date.

1.1 Vector Space

Intuitively a vector space is a space of "arrows". Vectors can be added and scaled.

More formally... for all vectors x, y, z and complex numbers a, b:

Definition: A set of vectors {e₁, ... ,e_n} forms a basis for the vector space if:

• For every vector x, there are numbers x¹, ... ,xⁿ such that x = x¹e₁ + ... + xⁿe_n
• There is no subset of {e₁, ... ,e_n} which satisfies the above condition.

The number of vectors required to form a basis for a vector space is the called the dimension of the vector space.

1.2 Linear Operators

A linear operator is a mapping A: V ® V such that A(ax + by) = a.Ax + b.Ay

1.3 Inner Products

The formal definition is significantly different from intuitive concept. Inner products allows us to think about "the angle between vectors" in the space. In particular

where ||.|| is the norm (length) of a vector. If the "angle" between vectors is 90º, they are said to be orthogonal. I.e. vectors are orthogonal if and only if <x|y> = 0 (cos q = 0, intuitively q = 90 degrees)

1.4 Norm

The norm of a vector corresponds to it's length. A inner product defines a norm by ||x|| = sqrt( <x|x> ). Formally, a norm has the following properties

1.5 Completeness

This is a technical requirement - which allows us to use calculus within the space. The space should not have "holes" in it, created by "missing" points. Formally, every cauchy sequence must converge. The requirement can be understood by looking at number sequences. Irrational numbers "plug" the holes in the real number line "between" fractions. If the distance between the numbers in a sequence becomes less and less, then the sequence must converges to a number in the space. E.g.

The sequence a_n consists of rational numbers and converges to e = 2.71828... but e is NOT a rational number. The space of rational numbers is therefore not complete. e is however a real number - the space of real numbers is complete.

1.6 Metric spaces

A metric space is equipped with a distance function, or metric, that gives the distance between any 2 points in the space. Formally, there exists a function, dist(.,.), such that

1.7      Hilbert Spaces

A Hilbert Space H is a vector space equipped with an inner product (denoted <.|.>), resulting norm (denoted ||.||) such that it is a complete metric space.

1.8      Dual Spaces

The dual space V^* of a vector space V is the vector space created by the linear functionals defined on V.
A linear functional is a mapping f: V ® F where F is the set of scalars (real or complex numbers) and f(ax + by) = a.f(x) + b.f(y).

V^* is isomorphic (looks like) V. If {e₁, ... ,e_n} forms a basis for V, then {e¹, ... ,eⁿ} forms a basis for V^* where eⁱ(e_k) = dⁱ_k

• V^* has the same dimension as V.
• Each vector x = S xⁱe_i is associated with a (co)vector x^* = S x_ieⁱ in V^*.
• If {x₁, ... ,x_N} is a basis for V, then {x^*₁, ... ,x^*_N} is a basis for V^*
• Vectors in V are called contravariant vectors, and vectors in V^* are called covariant vectors.

1.9      Tensor Spaces

Quantum Mechanics doesn't require a full formal treatment of Tensor Spaces; the definition given here is somewhat informal. The tensor product of 2 vector spaces V and W is a second vector space, denoted by V Ä W, constructed from V x W with addition and multiplication defined so

The tensor product is associative (the separate spaces are actually isomorphic).

• If V has dimension N and W has dimension M, then V Ä W has dimension N*M.
• If {x₁, ... ,x_N} is a basis for V, and {y₁, ... ,y_M} is a basis for W, then {x_i Ä y_k | i=1..N, k=1..M} forms a basis for V Ä W.

If x Ä y is an element of V Ä V^*, and the contraction of x Ä y is y(x).

Note for physicists

Traditionally physicists have defined tensors in terms of how the components of a tensor transform with a change of basis; the indices of components of (contravariant) vectors in V are denoted as superscripts, while indices of components of (covariant) vectors in V* are denoted as subscripts. For example, if x = S xⁱe_i Î V, and y = S y_ke^k Î V*, the tensor product x Ä y = S xⁱe_i Ä S y_ke^k = S S xⁱ y_k e_iÄ e^k would be denoted xⁱ y_k.

If x = S xⁱe_i Î V, and y = S y_ke^k Î V*, then the contraction of x Ä y = S S xⁱ y_k e^k(e_i) = S xⁱ y_i would be denoted xⁱ y_i.

The mathematical notation is much more powerful, and we will use the coordinate approach sparingly

1.10    Further theorems

Definition: A hermitian operator A is a linear operator such that A = A* where A* is the transpose of the complex conjugate.

• If A is a hermitian operator then <Y|Aj>. = <j|AY>*
• The eigenvalues of a hermitian operator are real (not complex).
• The eigenvectors of a hermitian operator are orthogonal.

Definition: A unitary operator U is a linear operator with the property UU* = 1 = U*U

• If U is a unitary operator then <UY|UY> = <Y|Y>.

if A&B are linear operators, then [A, B] = 0 if and only if A&B have the same eignevectors.

2.0     Standard Postulates of Quantum Mechanics

2.1   Postulates

The standard postulates of QM originate primarily from John Von Neumann's book Mathematical Foundations of Quantum Mechanics. It's still in print, and well worth the read.

Interpretations of QM have to be able to "explain" these postulates, or derive them if an alternative set of postulates are used.

(1) The possible states of a system are described by a Hilbert space; at any instant the state of the system will be described by a point in that Hilbert Space.

Why a Hilbert Space? A Hilbert Space H is a vector space equipped with an inner product (denoted <.|.>), resulting norm (denoted ||.||) such that it is a complete metric space. That raises the questions:

1. Why a vector space? A very, very good question.
2. Why require an inner product?
3. Why require a norm?
4. Why require completeness? - Allows us to use calculus within the space.
5. Why a metric space? - Allows us to use calculus within the space.

(2) An observable (physically measurable quantity A) corresponds to a linear hermitian operator (typically denoted A)on H.

1. Why should operators be linear? Another good question
2. Why hermitian? The requirement that the operator is hermitian ensures that measurements always return real numbers (See below).

(3) If a system is in a state Y, then the expected value of an observable A is given by

Y is an eignevector of A if

Why are measured states eignevectors and measured values eigenvalues? There is no mystery here: In general, a measurement of an observable A on a system in a state Y results in 2 things: (1) the system is described by a new state F, where (2) F is associated with the measured value A=a. More formally,

Furthermore, if the initial state Y is associated with a measured value A=a, then MeasurementOf(A, Y) ® Y The use of eignvalues and eigenvectors is simply a elegant way of associated a value with a vector (provided we assume the previous algebraic description).

2.2    Consequences

1. AY = aY Þ a measurement of quantity A will yield the result a with probabaility = 1 (not quite the same as certainty).
2. .. all of quantum mechanics...

2.3    Notation

Bra-ket notation

Dirac Bra-ket notation. Vectors in H looks lilke |x>. Vectors in H ^* look like by <y| .

The tensor product of a H x H ^* looks like |x><y|. The tensor product of a H ^* x H looks like <y|x>, which of course becomes notationally indistinguishable from the inner product <y|x> created by the natural contraction.

Bracket notation is particularly useful when describing states which are not particularly mathematical in nature. E.g |cat is dead >.

3.0      Heisenberg Uncertainty Principle

In 1929, Howard Robertson showed that if A, B and C are operators, [A,B] = iC, s_A is the standard deviation of expected distribution of measurements of A, then

In the case where A = x (a position operator) and B = p (a momentum operator), [A,B] = [x,p] = ih/2 p. Robertson's theorem implies

which is Heisenbegr's Uncertainty Principle. The proof is sufficently illuminating that it is reproduced.

Cauchy-Schwartz Inequality: For any vector Y, <AY|BY>² <= ||AY||² ||BY||²

|<AY|BY>|  ³  | im(<AY|BY>) |² = | 1/2(<AY|BY> - <AY|BY>*) =  1/2 |<AY|BY> - <BY|AY>| = 1/2 |<ABY|Y> - <BAY|Y>|  = 1/2 |<(AB -  BA)Y|Y>| = 1/2 E_Y([A,B])

Þ ||AY||² ||BY||² ³ <AY|BY>²  ³ 1/4 E_Y([A,B])²

Make the substitution A -> A - <A> and B -> B - <B>, then ||AY||² =s_A,Y

Þs_A,Ys_B,Y ³ 1/2 | E_Y([A,B]) |

4.0      Pure and Mixed States

A point Y in this space represents the best description possible of the system, and is sometimes called a "pure" state.

If the system could be in a number of possible (pure) states Y₁, Y₂, ... then there is added uncertainty and the system is described as being in a "mixed" state.

TODO: Origin of mixed states - e.g. particle sources, interaction of systems.

4.1      Mixed States

A Mixed state generally cannot be represented by a point in the state space, i.e. as a pure state.

Proof: Demonstrate by example. A electron source produces electrons with completely random spins. If a measurement is made, it is equally likely that the electron will be found in the spin up or down states, or spin right or spin left states. I.e. P(Y_u) = P(Y_d) = P(Y_r) = P(Y_l) = 0.5

Is the system in a well defined state? Suppose the system is in a pure state Y = a Y_u + b Y_d , then

Similarly P(Y_d) = 0.5 so b = ±(1/Ö2) and therefore Y = (1/Ö2) Y_u ± (1/Ö2) Y_d. I.e. Y = ±Y_r or Y = ±Y_l.
But P(Y_r|Y) = 0.5 so either P(Y_r|Y) = 0.5 = P(Y_r|±Y_r) = 1.0 which is nonsense, or P(Y_r|Y) = 0.5 = P(Y_r|±Y_l) = 0 which is also nonsense. The electron therefore cannot be described by a state vector Y.

Furthermore, once probability has been assigned to a basis, it has been assigned for each point in the space. I.e. It is no longer possible to independently assign probability to other points in the state space. In our example, once probability is assign to Y_u and Y_d, then it is also assigned to Y_r and Y_l, and any other point in the space.

4.2      The Density Operator

The density operator is the quantum version of a probability density function defined on the state space. Define the density operator r to be a member of H Ä H ^* such that

• For a pure state Y,   r = Y Ä Y^*. (It is often convenient to write r(x) = Y <Y|x>)
• For a mixed state, where the system has probability p_i of being in state |Y_i>,
  r = å p_iY_i Ä Y_i^*
  

Theorem: The probability that a system in a mixed state with be found to be in state Y is then given by

Theorem:

How do we tell if a density operator represents a pure state or not?
Theorem: r.r = r if and only if r = Y Ä Y^* for some Y.

Proof: Let r = Y Ä Y^*, then r.r = Y Ä Y^*( Y Ä Y^*) = Y Ä Y^* = r
Let r = å p_iY_i Ä Y_i^*, then r.r = å p_iY_i Ä Y_i^* ( å p_kY_k Ä Y_k^* ) = å å p_ip_k Y_i Ä Y_i^* ( Y_k Ä Y_k^*) = å å p_ip_i Y_i Ä Y_i^*
r.r = r Þ p_ip_i = p_i. Hence p_i = 0 or p_i = 1. Since å p_i = 1, there can be at most one p_i = 1. Therefore r = Y Ä_i Y_i^* for at most one i. QED.

4.3      Entanglement

Example 1: Independent particles

Example 2: Entangled particles (EPR)

The density operator r is invariant under the transformation d®u, u®d.

5.0      Information

Definition (Classical): The amount of information conveyed by a symbol is log₂(p) bits where p is the probability that he symbol will occur. The entropy of random variable is -åp_ilog₂(p_i)

TODO: This page needs to be completely restructured.

Quantum Interpretations

6.0      Spin

S-G apparatus (follow Feynman).

Physics Background: Special Relativity, Maxwells Equations - links only.

7.0      Quantum Field Theory

TODO: Lagrangian, U(1), SO(3), Poncaire, SU(2).

Copyright (c) Shaun O'Kane, 2003, 2004. You are free to redistribute this work provided you give due credit to it's author.