©Fernando Caracena 2013
A previous blog on the wave particle duality (W) presents several relations between particle properties (energy and momentum) and wave properties (frequency and wave length, respectively)
εν = hν (W.1a)
p = h / λ, (W.5)
which were proposed by Luis de Broglie for all subtomic particles of physics.
Alternatively, the above equations are written as
E = ħ ω (W.6a)
and
p = ħ k . (W.6b)
The above equations state a connection between what are normally interpreted as particle properties (E and p) and wave properties (ω and k); however there is some kind of dynamical connection not presented there. That is the problem that Erwin Schrödinger addressed in coming up with his famous wave equation. I do not know what was going on in his mind then, but what is presented below is a motivational background for his equation.
A simple equation describes a complex, one-dimensional wave travelling in the positive direction along the x-axis
f(x,t) = A ei(kx-ωt). (1a)
Using (W.6a and b) the above equation (1a) is rewritten as
f(x,t) = A ei[(p/ħ)x-(E/ħ)t]. (1b)
The question is, how do we extract energy and momentum from (1b)? A simple way is by differentiation
-iħ∂/∂x f(x,t) = p f(x,t) (2a)
iħ∂/∂t f(x,t) =E f(x,t) . (2b)
In (2a and b) there are two differential operators, which can be defined as follows, in bold characters:
Px = -iħ∂/∂x (3a)
and
E = iħ∂/∂t . (3b)
Rewriting (2 a and b), we have what is called the eigenvalue problem in its most general form:
Pxf(x,t) =px f(x,t) (4a)
E f(x,t) = E f(x,t) . (4b)
The real number p is the eigenvalue of the operator, Px; and E, that of E. The operators (Px and E) acting on a function [f(x,t)] result in the simple multiplication of that function by ordinary, real numbers (px and E respectively). The operators in (41) and (4b), which have real eignevalues, are called Hermitian. These operators are equal to their own transposed, complex conjugates, which are called Hermitian adjoints
E = E†
Px = Px† . (4d)
The Hamiltonian
A lot of work in classical mechanics just prior to the advent of modern physics went into the development of mechanics in very elegant form, one of which was called Hamiltonian dynamics, in which Newtonian mechanics was developed using a function (H) that represented the energy of a system of particles as a function of positions (qi) and mementa (pi ) of the particles (i=1,2,3..). The Hamiltonian is generally of the form
H=∑i [pi2/2mi+Vi(q1,q2,q3,....)], (5a)
where Viis the potential of theith particle produced on it by the configuration of the rest of the particles of the system.
Schrödinger hypothesized that a non-relativistic particle (travelling much slower than the speed of light) would satisfy the following wave equation
iħ∂/∂t ψ(r, t) = H ψ(r, t), (5b)
where H is the Hamiltonian operator formed by writing the appropriate variables as operators
H=P2/2mi+V(r). (4b)
Note that taking the Hermitian adjoint reverses the order of the operators, and they are take to act on the adjoint of the wave function to the left, instead of to the right
ψ†(r, t)(-iħ∂/∂t) = ψ†(r, t) H. (5c)
Strictly speaking, in this case it is not necessary to reverse the order of operators, since the wave function is an uncomplicated complex function of the temporal and spacial variables, and the operators are derivative operators. However, quantum mechanics in its most general form necessarily uses that above conventions.
"Where did we get that (equation) from? Nowhere. It is not possible to derive it from anything you know. It came out of the mind of Schrödinger."—Feynman
What is ψ ?
It is called the wave function, but originally, some dispute hang about it as to what it really was. Schrödinger thought that it represented some kind of charge amplitude. There is the Copenhagen Interpretation that most physicists subscribe to: it is a probability amplitude, the absolute square of which represents the probability of finding the subject particle at a given location at a given time. Note then that the Schrödinger Equation does not prescribe a path for a particle, but only the evolving probability amplitude of finding a particle throughout a region of space as a function of time time.
Since the wave function is complex, we define the absolute square of it, which is a real function, using a symbol that suggests a density
ρ(r,t) = ψ†(r, t) ψ(r,t) (6a)
Taking its time derivative, we have
∂/∂t ρ(r,t) =[∂/∂t ψ†(r, t)] ψ(r,t) + ψ†(r, t) [∂/∂t ψ(r,t)] . (6b)
Using (5a) and leaving out function arguments for simplicity we can rewrite (6b) as
∂ρ/∂t =[-ψ†(r, t) H/iħ] ψ + ψ†(r, t) [H/iħ ψ] , (6c)
where the square brackets limit the range of action of the operator.
With some work, we can rewrite (6c) as a continuity equation:
∂ρ/∂t =[-(-ħ2 ψ†(r, t) ∇2 )/iħ] ψ + ψ†(r, t) [(-ħ2∇2 )/iħ ψ]; (6d)
∂ρ/∂t =iħ ∇•[ψ†(r, t) (∇ψ)-(ψ†(r, t)∇)ψ]/(2m) ; (6e)
∂ρ(r,t)/∂t + ∇•j(r,t) =0 ; (7a)
j = iħ/(2m)[(ψ†(r, t) ∇)ψ - ψ†(r, t) (∇ψ(r, t) )]. (7b)
The vector j is reminiscent of the electric current density which appeared in the treatment of Maxwell's Equations, (ME.8). In this case, (7a) suggests that the Schrödinger equation represents a dynamics that is associated with a conserved quantity, such as electronic charge. The Copenhagen interpretation is that what is conserved is the probability of locating the electron described by the Schrödinger equation. However, there is something very subtle at work here. The wave function obtained for a hydrogen atom is a cloud in three dimensions moving in the electrical potential of a proton, which it surrounds as a single structure. It has an integrity that is not the blur of a small speck in orbit. An electron moving in orbit about the atomic nucleus is in a state that corresponds to a cloud shaped in space, which disappears when the electron is knocked out of that state of motion.
A future blog solves the Schrödinger equation for the harmonic oscillator using the method of operators and commutation relations.