©Fernando Caracena 2013
Mathematics is an experimental science, and definitions do not come first, but later on.
A parallel, complimentary treatment of this material is presented by Prof. Leonad Susskind at Stanford University in "Lecture 3 |Modern Physics: Quantum Mechanics." The differences between his approach and mine, could be profitable for the reader to consider because both treatments begin from opposite ends and agree in the middle. The treatment presented below, begins from intuitive principles and abstract mathematics, which is constructed experimentally, and shown to be self-consistent. Pro. Susskind begins from standard, recognized mathematics using discrete elements and approaches the more abstract principles from there.
Operators and state vectors
The development of quantum theory led to the development of a lot of interesting mathematics of operators, particularly by Dirac, who drew on the work of Oliver Heaviside. Dirac introduced a generalization of the Kronecker-delta of discrete indices, as one of continuous indices in the Dirac-delta function, which is related to Heaviside's step-function. At first, mathematicians objected to Dirac's work, but later became comfortable with it. It is very intuitive mathematics—although far out, it works!
In what follows, we shall develop the operator mathematics of Dirac, in an intuitive way for a single particle in one spatial dimension. The multi-particle system in three or more dimensions is handled in a similar way if one takes care to label everything properly and handle each dimension as it is here.
Our treatment of the mathematics is the experimental approach of Heaviside, which emphasizes intuition, but also shows that the ideas work and make sense. A mathematician would probably want to construct a series of existence theorems for this body of mathematics. What we have tried to do here is present the mathematics through an intuitive approach. The result is a usable mathematical tool kit that can serve in a number of calculations in advanced quantum mechanics.
Representations and transformations
State vectors of a particle moving in one dimension can be represented in terms of either of two conjugate variables (x or p), and there are mathematical transformations that can take us from one representation to the other. For example, in representing a state in spatial coordinates we project it on the state vector ket, |x>, the meaning of which is that the particle is at the location x. If a particle is detected there, then it cannot be anywhere else at the same time, which is represented by the probability amplitude
<x'|x> = 0, if x' ≠ x; (1a)
but if you search for it everywhere, you will find that it was indeed detected at x
∫-∞∞<x'|x> dx' =1. (1b)
The Dirac delta function has the required property to represent the scalar product,
<x'|x> =δ(x-x'). (1c)
δ(x-x') = 0 if x ≠ x' and ∞ if x = x', (1d)
∫-∞∞ dx' δ(x-x') = 1. (1e)
Also notice that any integral of the product of a function with the Dirac-delta function, extended over the full range of the variable of integration, evaluates the function at the point where the delta function is not zero,
f(x) = ∫-∞∞ dx' f(x') δ(x-x'). (1f)
A similar statement can be made about the momentum of a particle, it can be in only one momentum state at a time.
<p'|p> = 0, if p' ≠ p; (2a)
<p'|p> = δ(p-p'). (2b)
The state of a particle in general can be represented by a superposition of particle states of a single particle located a any point over any spatial range, by a distribution that is specified by a complex function ψ(x') of the x' coordinate, which is a dummy variable (i.e., you can represent it by any symbol so long as it is uniformly replaced)
|ψ> = ∫-∞∞ dx' ψ(x') |x'>. (3a)
This function can be extracted by a scalar product of a bra state vector <x| with the complex state vector | ψ>
<x|ψ> = ∫-∞∞ dx' ψ(x') <x|x'>,
which in couple of steps
<x|ψ> = ∫-∞∞ dx' ψ(x') δ(x-x')
<x|ψ> = ψ(x). (3b)
These operations (3a) through (3b) show that the procedures are reversible, so that we can begin with an unknown state vector and recover the complex function that would construct that vector.
As in ordinary vector analysis, we can define an identity operator, which allows us to go back and forth between spatial and momentum representations
I = ∫-∞∞ dx'|x'><x'| (4a)
I = ∫-∞∞ dp'|p'><p'|. (4b)
We get no inconsistency in applying (4a) to the state vector
I|ψ> =∫-∞∞ dx'|x'><x'|ψ>, (4c)
which reproduces (3a)
I|ψ> =∫-∞∞ dx' ψ(x')|x'>. (4d)
The mathematics of quantum mechanics is deigned to be able to carry the wave-particle duality, by design. In mathematics there is a similar duality between the form of a signal and its spectral content, which is central to communication theory. In one dimension, the spectral content can be represented by the wave number, which for a signal in more than one dimension, is just the component of a wave vector, in this case, in the x-direction
k = 2π/λ, (5a)
where λ is the wavelength of the particular harmonic component. One goes back and forth from spatial representation to spectral representations using Fourier Integrals
Φ(k) = (2π)-½ ∫-∞∞ dx' e-ikx'ψ(x') (5b)
ψ(x) = (2π)-½ ∫-∞∞ dk eikx Φ(k), (5c)
which are as reversible as the representations of quantum theory. Notice that the factor, (2π)-½ e-ikx' ,involved in (5b) is the complex conjugate of that involved in (5c).
Fourier analysis is an important tool in solving many real-world problems in both science and engineering. Fourier analysis over a limited interval of space results in a discrete series of components with complex coefficients. Analysis using Fourier integrals allows one to define a continuous function over a limited domain in terms of an infinite number of harmonic components that span spectral space over a continuous range.
The Fourier formalism is adapted to quantum mechanics by using the follow de Broglie relation
p = ħ k (= h/λ) (6a)
In modifying (5b) and (5c) we should spread out the normalizing factor ħ-1
symmetrically along with the factor of 2π, as follows:
Φ(p) = (2π ħ)-½ ∫-∞∞ dx' e-ip/ ħx'ψ(x') (6b)
ψ(x) = (2π ħ)-½ ∫-∞∞ dp eip/ ħx Φ(p). (6c)
We can associate these transformed functions with the representation of a state vector projected alternatively into momentum and coordinate representation, respectively,
Φ(p) =<p|ψ> (6d)
ψ(x) =<x|ψ>. (6e)
Using the identity operator we can write transforms between the two representations as follows:
<p|Φ> =∫-∞∞ dx' <p|x'><x'|ψ> (6f)
Φ(p) =∫-∞∞ dx' <p|x'><x'|ψ> . (6g)
A scalar product in (6g) can be identified by comparing this equation with (6b)
<p|x'> = (2π ħ)-½ e-ip/ ħx', (7a)
which through its Hermitian adjoint yields the corresponding factor in (6c)
<p|x'>† = <x'|p>,
<x'|p> = (2π ħ)-½ eip/ ħx'. (7b)
The above procedure can be also applied to the coordinate representation, resulting in the same factors as follows:
ψ(x) =<x|ψ>; (7c)
ψ(x) =∫-∞∞ dp <x|p><p|ψ>; (7d)
using (7b) we can write (7d) as
ψ(x) = (2π ħ)-½ ∫-∞∞ dp eip/ ħx Φ(p), (7c)
which is identical to (6c).
In the momentum representation, the momentum operator P has an eigenvalue p
P|p> = p|p>, (8a)
which is a real number. If we take the scalar product of both sides of (8a) with the bra vector <x| we get the following:
<x|P|p> = p <x|p>,
which according to (7b) evaluates as follows
<x|P|p> = (2π ħ)-½ p eip/ ħx. (8b)
We recognize that (8b) results from taking the derivative of the exponential
<x|P|p> = (2π ħ)-½(-iħ) ∂eip/ ħx/∂x,
which using (7b) can be put back as
<x|P|p> = -iħ ∂<x|p>/∂x. (8c)
In the coordinate representation, the momentum operator becomes the differential operator
P → -iħ ∂/∂x
Using (7d), show that the momentum operator becomes a differential operator acting as follows on the wave function ψ(x):
<x|P|ψ> = -iħ ∂ψ(x)/∂x. (8d)
It is left to the reader to evaluate the position operator X in the momentum representation. Hint, start from the following equation
X|x> = x|x>. (9)
Five postulates of quantum mechanics
In his Lecture 3 on Quantum Mechanics, Prof. Susskind presents five postulates for the mathematics of quantum mechanics, which we briefly summarize as follows:
- The states of quantum mechanical systems are represented by complex vectors [in a Hilbert space.]
- Observables (quantities capable of being experimentally measured) correspond to the collection of Hermitian operators,
- which have real eigenvalues that specify what you can measure.
- The state vectors for which a hermitian operator has definite eigenvalues are the eigenvectors of that operator.
- The eigenvectorsof a Hermitian operator form an orthonormal set of vectors,
|λn>, and the probability of an arbitrary ket's having anyone of that set is given by
P(λm) = |<λm|A>|2.
He states that the postulates could be reduced to a smaller number, but that the five postulates give a more transparent definition of the mathematics of quantum states and observables.