Download Schrödinger equation (Text 5.3)

Schrödinger equation (Text 5.3)
In classical mechanics, conservation of energy :
p2
E = KE + PE =
+ U( x)
2m
In quantum mechanics, in operator form :
p̂ 2
Ĥ =
+ U(x, t)
2m
∂
h2 ∂2
⇒ ih
=+ U(x, t)
2
∂t
2m ∂x
Apply these operators to the wave function, we get the Schrödinger equation:
∂
h2 ∂2
ih Ψ(x, t) = Ψ(x, t) + U(x, t)Ψ(x, t)
2
2m ∂x
∂t
Schrödinger equation (Text 5.3)
Example. In free space, U(x,t)=0 and the Schrödinger equation becomes
∂
h2 ∂2
ih Ψ (x, t) = Ψ (x, t)
2
2m ∂x
∂t
Plane wave Ψ(x,t) = Aei(kx-ωt) is a solution of this equation:
Ψ (x, t) = Aei(kx -ωt)
∂
∂
Ψ (x, t) = ih Aei(kx -ωt) = hωAei(kx -ωt)
∂t
∂t
h2 ∂2
h2 ∂2
Ψ (x, t) = Aei(kx -ωt)
2
2
2m ∂x
2m ∂x
ikh 2 ∂
=Aei(kx -ωt)
2m ∂x
h 2 k 2 i(kx -ωt)
=
Ae
2m
h 2k 2
Dispersion relation for
i(kx -ωt)
∴ Ae
is a solution if
= hω
“free particle”
2m
ih
Schrödinger equation (Text 5.3)
Schrödinger equation cannot be derived from
other basic principle of physics, it is a basic
principle in itself.
In reverse, if we accept Schrödinger equation as a basic principle, then the
classical Newton’s law of motion can be derived from Schrödinger equation
provided the classical interpretation of physical quantities is understood to be
the expectation value of the wave function.
Time independent Schrödinger equation (Text
5.7)
When an operator acts on a wavefunction, you will have
another function. For4 example,
If ψ (x) = x
∂ 4
p̂ψ = - ih x =
-1
i2
x3
h3
∂x
Another wavefunction
Philosophically speaking, this is the case because the
measurement (i.e. operator) has changed the system (wave
function).
Time independent Schrödinger equation (Text
5.7)
Some function may be “special” to an operator in this way:
Ĝ ψ n ( x) = G n ψ n ( x)
(Equation 5.34 of text)
in which Gn is a number. If this happens, ψn(x) is known as
the eigenfunction, or eigenstate, or eigenvector of operator G.
Gn is known as the eigenvalue of operator. The subscript n is
used to label or name the function, because there can be many
eigenfucntions for the same operator.
Philosophically speaking, eigenfunction represents special
state that the corresponding physical quantity (operator) can
be precisely measured without disturbing system.
Time independent Schrödinger equation (Text
5.7)
More about eigenfunction and eigenvalue:
1.
Different operators have different sets of eigenfunctions.
2.
For the quantity to be physical, the eigenvalues Gn have to be real (i.e.
not a complex number).
3.
If two different operators A and B share the same set of
eigenfunctions, the two operators commute because their
measurements will not affect the system.
AB=BA
4. For example, plane wave Ψ(x,t) = ei(kx-ωt) is an eigenfunction of
momentum:
∂
p̂ Ψ (x, t) = - ih ei(kx -ωt) = hkei(kx -ωt) = hkΨ (x, t)
∂x
hk is the eigenvalue.