Schrodingers Equation and Observables

Introduction

In Quantum mechanics, all observables - measurables variables - are replaced with operators. This is one of several fundamental differences between classical, continuous mechanics and quantum mechanics. Examples of such observables include, position, momentum, spin, and of central importance to chemistry, the energy. It is worth at this point to become familiar with linear operators and eigenvalue-equations. In brief, an operator is a mathematical construct which transforms one function into another; eigenvalue-equations are those which upon application of the operation, return the original function along with a constant. Linear operators are those which can be applied distributively to individual components of a sum of functions. Consider the differentiation operator which is linear, and compare that to trignonometric functions which are non-linear, ie., representing the differentiation operator by `hatD`, and where `F(x)=f(x)+g(x)`, then

`hatDF(x)=hatDf(x)+hatDg(x)`
which is linear

compare this to the sin function,

`sin(F(x))=sin(f(x)+g(x))!=sin(f(x))+sin(g(x)`
which is non-linear

In order for the operator formulation of observables to work, we require that once we have applied the operation to the function, the function remains unchanged. This is embodied by the definition of an eigenvalue-equation, an example of which follows:

`hatAf(x)=lambdaf(x)`
an eigenvalue-equation

In such a formulation, `hatA` is the eigen-operator, f(x) is the eigenfunction, and `lambda` is the eigenvalue. Evidently, upon the operator acting upon the function, we obtain the function back (it appears on both sides of the equation), but along with a constant, `lambda`. Thus, `hatA` represents an observables-operator, and `lambda`, the observable, measurable expectation value.

The Position Observable

The position observable is the most simple of the observable operators,

`hatx=x`

Thus, the position operator `hatx` is simply the wave-function multiplied by x.

The Potential Energy Observable

The potential energy observable is obtain by applying the potential energy operator. The form of this operator depends on the particular scenario in which the particle is bound, and therefore on the external forces acting upon the particle. Like the position operator, the potential energy operator simply multiplies the wavefunction by the potential energy.

`hatV=V`

The Momentum Observable

As demonstrated for a de Broglie wave, the wave-number, k, is related to the momentum of the particle through the expression,

`p=hbark`

This means, that for the de Broglie (and all valid solutions to the wave-equation), when the momentum operator acts upon a wavefunction, it must extract an eigenvalue equal to `hbark`. Upon a reinspection of the de Broglie wavefunction,

`Psi(x,t)=Aexp(i(kx-omegat+phi)))`

We see that upon partial differentiation with respect to x, we obtain, a function of k. To do this, we apply the chain rule,

`del/(delx) Aexp(i(kx-omegat+phi)) = ik Aexp(i(kx-omegat+phi))`

Were differentiation to underpin the momentum operator, we require that upon its action we obtain the eigenvalue of momentum `hbark`. Firstly we can see that the above equation is an eigenvalue-equation - the eigenfunction is present on both sides, and we have indeed obained an eigenvalue, just not the one we want. We can however convert an eigenvalue of `ik` to `hbark` by multiplying by `-ihbar`. Thus, we can define the momentum operator as,

`hatp=-ihbar del/(delx)`

The Energy Observable

In classical mechanics, and in particular, Hamiltonian mechanics, the Hamiltonian function is one representing the total energy of a system, both kinetic and potential. This is expressed as,

`H=T+V`
where H is the Hamiltonian, T is the kinetic energy and V is the potential energy

The quantum mechanical equivalent must involve operators. Hence we replace the Hamiltonian function with the Hamiltonian operator, `hatH`. Again referring to the de Broglie wavefunction, we recall that the energy of the particle is equal `hbaromega`. Thus we require the Hamiltonian Operator to extract an eigenvalue equal to `hbaromega`. Inspecting the de Broglie wavefunction, we see that partial differentiation with respect to time extracts the eigenvalue `omega`,

`del/(delt) Aexp(i(kx-omegat+phi)) = -iomega Aexp(i(kx-omegat+phi))`

Following the same process as that for the momentum operator, we can convert this eigenvalue to energy by multiplying by `ihbar`. Thus we can formulate the Hamiltonian operator as follows,

`hatH=ihbar del/(delt)`

However, through the relationship between momentum and kinetic energy, viz.,

`E_k=p^2/(2m)`

...we can go a little further and also express the Hamiltonian in terms of the momentum operator, namely,

`hatH=hatp^2/(2m)`

Using our definition for `hatp` this becomes,

`hatH=-hbar^2/(2m) del^2/(delx^2)`

Curiously, this leads to the scenario in which the two partial derivatives, one with respect to t and the other with respect to x, are both equal. In order for this to be the case, both partial derivatives must be equal a constant, and this as we realise, is the energy expectation value. Thus,

`hatH=ihbar del/(delt)=-hbar^2/(2m) del^2/(delx^2)`

We note that since the de Broglie wavefunction is one for a free particle, there are no external forces acting on the particle and so the potential energy is zero. However, to generalise the Hamilitonian operator to bound particles (ie., those experiencing external forces), like with the Hamiltonian function, we must add a potential energy term, thus,

`ihbar del/(delt)=hatH=-hbar^2/(2m) del^2/(delx^2)+hatV(x)`

Thus we arrive at an eigenvalue-equation representing the total energy of the system,

`hatHPsi(x,t)=EPsi(x,t)`

where E is an eigen-value of the Hamilitonian operator of the eigenfunction, `Psi(x,t)`

This leads onto the definition of the Schrodinger wave-equation,

`ihbar(del)/(delt)Psi(x,t)=(-hbar^2/(2m)(del^2)/(delx^2)+hatV(x)) Psi(x,t)`

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