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The Mathematical Foundations
of Quantum Mechanics
in a Special Hilbert Space

This material is to present the mathematical foundations of quantum mechanics but with an emphasis on the physical significance of the mathematics. The mathematical foundations of quantum mechanics were presented a long time ago in a full book by John von Neumann in which he stressed achieving mathematical rigor. But a theory may be mathematically rigorous yet physically irrelevant.

Hilbert Spaces

A Hilbert space is a generalization of vector spaces that allows for infinite dimensionality. Another name for Hilbert spaces is Cauchy-Complete Inner Product Spaces. Any vector space is a Hilbert space.

Cauchy-Complete means any Cauchy convergent sequence converges to an element in the space.

The relevant Hilbert Space for quantum mechanics is an infinite dimensional vector space. An infinite dimensional vector space is not so esoteric as it sounds. Its elements are merely functions from a real interval to a continuum set. For quantum mechanics the real interval is usually [−∞, ∞]. The continuum is usually the set of complex numbers.

There is a special set of functions from the real numbers to the complex numbers satisfying a certain condition. That condition is that

−∞ |f(x)|²dx = 1

where |f(x)|² is equal to f(x)*f(x), f(x)* being the complex conjugate of f(x). Such functions are called wave functions and |f(x)|² is the probabity density. A wave function is in the nature of a state of the physical system.

The nature and the physical significance of a complex value is problematical. Its squared magnitude, being a probability density, is physically significant. But the wave function is not just the square root of probability density. Any complex number on a circle of a radius equal to the positive square root of probability density satisfies the physically observable conditions. The value of the wave function can only come from the solution to a Schrödinger equation.

A Hilbert space being an inner product space means that the inner product of any two elements f and g the inner product I(f, g) is defined; i.e.,

I(f, g) = ∫−∞f(x)*g(x)dx

This definition requires that the integral converge to a finite complex number.

Thus for any wave function f in the special set

I(f, f) =1

Expected Values

Let f(x) be any element of the Hilbert space and A(x) any observable. Then the expected value of A given the physical system is in the state f is

E{A: f} = ∫−∞A(x)|f(x)|²dx
= ∫−∞f(x)*A(x)f(x)dx = I(f, Af) = I(A*f, f)

So far all of this is well and good. It is straight forward mathematics. There is however the problem of what the inner product I(f, g) represents if f≠g. That willbe dealt with later.

However, the presentations on this topic go on to define operators and their expected values.

Observables and their Operators

For simple physical systems involving a particle any observable quantity Q is a function of its momentum p and and its location x; i.e.,

Q = Q(p, x)

For example, kinetic energy K is equal to p²/(2m). For a harmonic oscillator the potential energy V is equal to ½kx².

The operator Q^ for an observable Q(p, x) is formed by replacing p with (h/i)(d/dx) and x is left as x. Any exponent of p is converted into the order of the differentiation by x. Thus p² → (∂²/∂x²).

Determinate States

A state does not mean a static condition; instead it means a repetitive situation. A rapidly rotating fan can be contrued as an unchanging translucent disk. But this is not the reality and should not be taken as evidence of the immateriality of the fan.

Consider this model of measurement. A physical system is in state Ψ, The measurement of an observable Q^ produces a value r and a shift of the system to the state Φ; i.e.,

Q^Ψ = rΦ

A determinate state is one in which measurement produces a value but no change in the state of the system; i.e.,

Q^Ψ = qΨ

Thus the determinate states of a physical system for an observable Q are just the eigenfunctions of the operator Q^.

The Expected Value of Momentum
for a Harmonic Oscillator

In this case Q=p and Q^=(h/i)(d/dx).

Let Φ be an eigenfunction of p^. It satisfies the equation

(h/i)dΦ/dx = pΦ

It has the solution

Φ(x) = (1/(2πh)½)exp(ipx/h))

The inner product of this solution with the solution for another value of momentum p' is give by

I(Φ, Φ') = δ(p−p')

where δ(z) is Dirac's delta function.

(To be continued.)


Sources:

David J. Griffiths, Introduction to Quantun Mechanics, Cambridge University Press, (2nd edition) 2017.

John von Neumann, The Mathematical Foundations of Quantum Mechanics, Princeton University Press, Princeton, N.J., 1955.


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