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differential equations. We review it here in order to highlight the aspects that
will be abstracted later on.
4.1.1 Separation of space and time
If the potential is time-independent, i.e. if V(x,t) = V(x), the Schr¨odinger
equation can be simplified by separating the variables
ψ(x,t)= u (x)f (t). (4.7)
n n
Xn
This is an ansatz for the solution, which can be justified referring to gen-
eral theorems on partial differential equations. [35]. Here, u and f are
n n
{ } { }
enumerable infinite sets of functions.
If the ansatz is inserted into the Schr¨odinger equation one gets
∂f(t) ¯h2 ∂2u(x)
i¯hu(x) = f(t) +V(x)u(x)f(t),
∂t −2m ∂x2
which upon division by u(x)f(t) yields
1∂f 1 ¯h2 ∂2u
i¯h = [ +Vu].
f ∂t u −2m∂x2
Now,sincethelefthandsideisindependentofxandtherighthandsideisin-
dependentoft,bothsidesmustbeequaltothesameconstantE. Thisconstant
is calledaseparation constant. We thus gettwoordinarydifferentialequations,
oneforthetime-dependentfunctionf andoneforthespace-dependentfunction
u,
∂f
i¯h =Ef (4.8)
∂t
¯h2 ∂2u
+Vu=Eu. (4.9)
−2m∂x2
The first equation is easy to solve
iEt
f(t)=Cexp( ), (4.10)
¯h
where C is a constant.
The second equation is an eigenvalue equation of the Sturm-Liouville type,
and its general form can be written abstractly as
Hu=Eu.
The solutions to this equation will be precisely the functions u (x) in the
n
expansion (4.7). They are referred to as eigenfunctions and the constants E
n
as eigenvalues.
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Forspecialformsofthepotential V (andappropriateboundaryconditions),
theequationdefineswell-knownsystemsoforthonormalfunctions u (x) with
m
{ }
the index mrunning oversome infinite subsetofZ. The generalsolutionto the
wave equation can then be written by inserting (4.10) into (4.7)
ψ(x,t)= u (x)exp(iE t/¯h),
n n
Xn
where the constantC has been hidden in the as yet undetermined functions
u .
n
4.1.2 Particle in a potential well
With the general groundwork done, we will now turn to our first example,
a quantum particle trapped within a container with impenetrable walls. In,
reality there is no such thing, but it can be mimicked by choosing a potential
of the form
+ , if x >a
V(x)= ∞ | | (4.11)
(cid:26)0, if x <a
| |
with the walls at the locations x = a and x = a. The impenetrability of
−
thewallsismodeledbytheinfinitevalueforthepotentialoutsidethewell. This
potential is often called a square well potential.
Sincethepotentialhasthreedistinctregions,beingdiscontinuousatx= a,
±
the equation must be solved in the three regions separately. However, since in
the two regionsx< a andx>a, the potential is infinite, the function u must
−
beequaltozerohere. Furthermore,uitselfmustbe continuousatthe potential
walls. This translates into boundary conditions for the solution in the region
x <a. Therefore, we get
| |
¯h2 ∂2
u=Eu (4.12)
− 2m∂x2
with boundary conditions
u(a)=u( a)=0. (4.13)
−
In this form, the boundary conditions are quite easy to understand. In
classicalphysics,noparticlecanpassfromasregionwithfinitepotentialenergy
into a region with infinite potential energy, as that would require an infinite
kinetic energy. This is true in quantum physics also. And since the u in some
sense corresponds to the presence of the particle (in a way that will explained
later), those boundary conditions corresponds to the impenetrable walls.
This differential equation has the well known solution
u(x)=Asinkx+Bcoskx
with k = 2mE/¯h2.
q
Inserting the boundary conditions, we get the two equations
72
Asinka+Bcoska=0
,