SCHRODINGER EQUATION
REVISION ON WAVE
Wave: can be defined as a disturbance that travels from one location to another carrying energy
along with it without transporting the matter. The disturbance goes forward as the particle
undergoes to and fro motion.(oscillation). Examples: a stone dropped in the water, sets up
circular ring of disturbance (up and down motion of water particles) which travels outward from
the point at which the stone landed in water. The disturbance is called water wave.
The above is an example of wave motion.
CHARACTERISTICS OF WAVES
Amplitude: this is the maximum displacement of a particle from the point of rest in meters (m)
Wavelength: this is the distance between two successive crests and troughs of the wave. It is
denoted by λ(lamda) and measured in meters (m)
Period: this is the time taken for one oscillation. It is denoted by T and measured in seconds.
T = 2π
w
w = 2πf
T = 2π =1
2πf f
Frequency:This is the number of cycles/oscillation/ vibration in one second.It is measured in
hertz(hz).
F=1
T
Phase Vibration: in-phase vibration exists at 2 points on a wave if those points undergo
vibration that are in same direction, in step
A C
Points A and C are said to vibrate in phase since they move up and down together while A and B
vibrate out of phase.
Velocity: The displacement through which the wave id propagated in one second. i.e v = fλ.
Representation of waves
y
y = Asin(wt-φ)
where w = 2πf; φ = kx and k = 2π
λ
y = Asin 2πft - 2πx
λ
The Schr¨odinger wave equation
We have noted in previous lectures that all particles, both light and matter, can be described as a
localised wave packet.
• De Broglie suggested a relationship between the effective wavelength of the wave function
associated with a given matter or light particle its the momentum. This relationship was
subsequently confirmed experimentally for electrons.
• Consideration of the two slit experiment has provided an understanding of what we can
and cannot achieve with the wave function representing the particle: The wave function Ψ
is not observable. According to the statistical interpretation of Born, the quantity Ψ ∗Ψ = |
Ψ2| is observable and represents the probability density of locating the particle in a given
elemental volume.
To understand the wave function further, we require a wave equation from which we can study
the evolution of wave functions as a function of position and time, in general within a potential
field (e.g. the potential fields associated with the Coulomb or strong nuclear force).
As we shall see, manipulation of the wave equation will permit us to calculate “most probable”
values of a particle’s position, momentum, energy, etc. These quantities form the study of
mechanics within classical physics. Our quantum theory has now become quantum mechanics
– the description of mechanical physics on the quantum scale. The particular sub-branch of
quantum mechanics accessible via wave theory is sometimes referred to as wave mechanics.
we investigated solutions to the classical wave equation of the form
Ψ(x,t) = Asin(kx − ωt + φ) (9)
where k is the wave number, ω is the angular frequency and φ is a phase constant. The wave is moving
in the positive x–direction. This “classical” wave is not a valid solution to the time– dependent
Schr¨odinger equation.
We start by considering the three derivative terms
. (10)
Inserting these relations into the time–dependent Schr¨odinger equation we obtain
, (11)
