Lecture 3: The Time Dependent Schrödinger Equation
The material in this lecture is not covered in Atkins. It is required to understand postulate 6 and 11.5 The informtion of a wavefunction
Lecture on-line The Time Dependent Schrödinger Equation (PDF) The time Dependent Schroedinger Equation (HTML) The time dependent Schrödinger Equation (PowerPoint) Tutorials on-line The postulates of quantum mechanics (This is the writeup for Dry-lab-II( This lecture coveres parts of postulate 6) Time Dependent Schrödinger Equation The Development of Classical Mechanics Experimental Background for Quantum mecahnics Early Development of Quantum mechanics Audio-visuals on-line review of the Schrödinger equation and the Born postulate (PDF) review of the Schrödinger equation and the Born postulate (HTML) review of Schrödinger equation and Born postulate (PowerPoint **, 1MB) Slides from the text book (From the CD included in Atkins ,**)
Consider a particle of mass m that is moving in onedimension. Let its position be given by x
O
X
Let the particle be subject to the potential V(x, t)
O
V
V(X,t1) V(X,t2)
All properties of such a particle is in quantum mechanics determined by the wavefunction Ψ( ,x ) t of the system
Time Dependent Schrödinger Equation setting up equation
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Time Dependent Schrödinger Equation
X
V(x,t) setting up equation
Time Dependent Schrödinger Equation
A system that changes with timeis described by the time-dependent Schrödinger equation
−
hiδΨ(x,t)
δt= ˆ H Ψ(x,t)
according to postulate 6
Where ˆ H is the Hamiltonian ofthe system:
ˆ H = −h2
2mδ2
δx2 +V(x,t)
for 1D-particle
setting up equation
−
hiδΨ(x,t)
δt=−
h2
2mδ2Ψ(x,t)
δx2 +V(x,t)Ψ(x,t)
The time dependent Schrödinger equation
The wavefunction Ψ( ,x )t is also referred to as
The statefunction
Our state will in general change with time due to V(x, t). Thus Ψ is a function of time and space
Time Dependent Schrödinger Equation setting up equation
The wavefunction does not have any physical interpretation.However :
P(x,t) = Ψ(x,t)Ψ(x,t)*dx
Probability density
ox
dx
Ψ( x, t)Ψ*( x, t)dx
Is the probability at time t to find the particle between x and x +Δ .x
Time Dependent Schrödinger Equation
will change with time
Probability from wavefunction
It is important to note that the particle is not distributed
over a large region as a charge cloud
It is the probability patterns (wave function) used to describe the electron motion that behaves like waves and satisfies a wave equation
Ψ(x,t)Ψ(x,t)*
Time Dependent Schrödinger Equation Probability from wavefunction
Consider a large number N of identical boxes with identical particles all described by the same wavefunction Ψ(x,t) :
Then :dnxN
=Ψ(x, t)Ψ* (x,t)dx
Let dnx denote the number of particlewhich at the same time is found between x and x +Δx
Time Dependent Schrödinger Equation Probability from wavefunction
−
hiδΨ(x, t)δt
= −h2
2mδ2Ψ(x, t)
δx2 +V(x, t)Ψ(x, t)
The time - dependent Schroedinger equation :
O
V
V(X)
Can be simplifiedin those cases wherethe potential V onlydepends on the position : V(t, x) - >V(x)
Time Dependent Schrödinger Equation with time independent potential energy
We might try to find a solution of the form :
Ψ(x, t) = f (t)ψ(x)We have
δΨ(x,t)
δt=δψ((x)f(t))
δt= ψ(x)
δf(t)
δt
δ2Ψ(x, t)
δx2 =δ2ψ((x)f(t))
δx2 = f(t)δ2ψ(x)
δx2
and
Time Dependent Schrödinger Equation with time independent potential energy:separation of time and space
−
hiψ(x)
δf(t)δt
=−h2
2mf(t)
δ2ψ(x)
δx2 +V(x)f(t)ψ(x)
A substitution of Ψ(x,t)= f(t)ψ(x)
into the Schrödinger equation thus affords:
Simplyfied Time Dependent Schrödinger Equation
−
hiδΨ(x, t)δt
= −h2
2mδ2Ψ(x, t)
δx2 +V(x, t)Ψ(x, t)
with time independent potential energy:separation of time and space
Simplyfied Time Dependent Schrödinger Equation
A multiplication from the left by1
f (t)ψ(x) affords:
−
hi
1f(t)
δf(t)δt
=−h2
2m1
ψ(x)δ2ψ(x)
δx2 +V(x)
The R.H.S. does not depend on t if we now assume that V is time independent. Thus, the L.H.S. must also be independent of t
−
hiψ(x)
δf(t)δt
=−h2
2mf(t)
δ2ψ(x)
δx2 +V(x)f(t)ψ(x)
with time independent potential energy:separation of time and space
−
hi
1
f (t)
δf(t)
δt= E = cons tan t
Thus :
The L.H.S. does not depend on x so the R.H.S. must also be independent of x and equal to the same constant, E.
−
h2
2m1
ψ(x)δ2ψ(x)
δx2 +V(x)=E =constant
Simplyfied Time Dependent Schrödinger Equation with time independent potential energy:separation of time and space
−
hi
1
f (t)
δf(t)
δt= E = cons tan t
We can now solve for f(t) :
Or :
δf(t)
f (t)= −
i
hEδt
Now integrating from time t=0 to t=to on both sides affords:
o
to∫δ ( )f tf (t)
=−o
to∫
ih
Eδt
Simplyfied Time Dependent Schrödinger Equation with time independent potential energy:separation of time and space
o
to∫δ ( )f tf (t)
=−o
to∫
ih
Eδt
ln[f(to)]−ln[f(o)]=−
ihE[to−0]
ln[ f (to )] =−
ih
Eto + [ln f (o)] Cons tan t
ln[ f (to )] =−
ih
Eto +C
Simplyfied Time Dependent Schrödinger Equation with time independent potential energy:separation of time and space
ln[ f (to )] =−
ih
Eto +COr:
f(t)=Exp−
ihEt+C
⎡ ⎣ ⎢
⎤ ⎦ ⎥
=ExpC[ ]Exp−ihEt
⎡ ⎣ ⎢
⎤ ⎦ ⎥
f (t) =ExpC[ ](cos
Eh
t ⎡ ⎣
⎤ ⎦−i sin
Eh
t ⎡ ⎣
⎤ ⎦)
Simplified Time Dependent Schrödinger Equation
with time independent potential energy:separation of time and space
with time independent potential energy:separation of time and space
Simplified Time Dependent Schrödinger Equation
Change of sign of f(t) with time
t=
π2(h/E)
+
t = (h/ E)
t=3π
2(h/E) t= (h/ E)
-i
- i +
t=0 π 2π
i
f (t) =ExpC[ ](cos
Eh
t ⎡ ⎣
⎤ ⎦−i sin
Eh
t ⎡ ⎣
⎤ ⎦)
−
h2
2m1
ψ(x)δ2ψ(x)
δx2 +V(x)=E
The equation for ψ(x) is given by
Simplified Time Dependent Schrödinger Equation
Time independent Schrödinger equation
with time independent potential energy:separation of time and space
−
h2
2mδ2ψ(x)
δx2 +ψ(x)V(x)=Eψ(x)
Or:
Simplified Time Dependent Schrödinger Equation
Time independent Schrödinger equation
with time independent potential energy:separation of time and space
−
h2
2mδ2ψ(x)
δx2 +ψ(x)V(x)=Eψ(x)
This is the time-independent Schroedinger Equation for a particle moving in the time independent potential V(x)
It is a postulate of Quantum Mechanics that E is the total energy of the system
Part of QM postulate 6
The total wavefunction for a one-dimentional particle in a potential V(x) is given by
Ψ(x, t) = f (t)ψ(x)
= Exp[C]Exp[−iEht]ψ(x)
= AExp[−iEht]ψ(x)
Simplified Time Dependent Schrödinger Equation
Time independent Schrödinger equation
with time independent potential energy:separation of time and space
If ψ( )x is a solution to
−
h2
2mδ2ψ(x)
δx2 +ψ(x)V(x)=Eψ(x)
So is Aψ( )x
−
h2
2mδ2(Aψ(x))
δx2 +Aψ(x)V(x)=AEψ(x)
Simplified Time Dependent Schrödinger Equation
Lecture 2
Time independent Schrödinger equation
with time independent potential energy:separation of time and space
or :
−
h2
2mδ2ψ'(x)
δx2 +ψ'(x)V(x)=Eψ'(x)
with ψ'( )x =Aψ( )x
−
h2
2mδ2(Aψ(x))
δx2 +Aψ(x)V(x)=AEψ(x)
Simplified Time Dependent Schrödinger Equation
time independent probability function
with time independent potential energy:separation of time and space
Thus we can write without loss of generality for a particle in a time-independent potential
Ψ(x, t) = Exp[−i
E
ht]ψ(x)
This wavefunction is time dependent and complex.
Let us now look at the corresponding probability density
Ψ(x, t)Ψ*(x, t)
Simplified Time Dependent Schrödinger Equation
time independent probability function
with time independent potential energy:separation of time and space
Ψ(x, t)Ψ*(x, t) = Exp[−iE
ht]ψ(x)
×Exp[iEht]ψ*(x) = ψ(x)ψ*(x)
We have :
Thus , states describing systems with a time-independent potential V(x) have a time-independent (stationary) probability density.
Simplified Time Dependent Schrödinger Equation
time independent probability function
with time independent potential energy:separation of time and space
Ψ(x, t)Ψ*(x, t) = Exp[−iE
ht]ψ(x)
×Exp[iEht]ψ*(x) = ψ(x)ψ*(x)
This does not imply that the particle is stationary.However, it means that the probability of finding a particle in the interval x + -1/2Δ x to x+1/2Δ x is
.constant
Simplified Time Dependent Schrödinger Equation stationary states
ψ(x)ψ*(x)dx Independent of time
We say that systems that can be described by wave functions of the type
Ψ(x, t) = Exp[−iEht]ψ(x)
Represent Stationary states
Simplified Time Dependent Schrödinger Equation stationary states
Postulate 6The time development of thestate of an undisturbed systemis given by the time-dependentSchrödinger equation
Simplified Time Dependent Schrödinger Equation
−
hiδΨ(x,t)
δt= ˆ H Ψ(x,t)
where ˆ H is the Hamiltonian(i.e. energy) operatorfor the quantum mechanical system
What you should know from this lecture
−
hiδΨ(x,t)
δt= ˆ H Ψ(x,t)
1. You should know postulate 6 and the form of thetime dependent Schrödinger equation
2. You should know that the wavefunction for systems where the potential energy is independent oftime [V(x,t) → V(x)] is given by
Ψ(x,t)=Exp[−iEh
t]ψ(x)
Where ψ(x) is a solution to the time-independentSchrödinger equation: Hψ(x)=Eψ(x),and E is the energy of the system.
What you should know from this lecture
3. Systems with a time independent potentialenergy [V(x,t) → V(x) ] have a time-independentprobability density:
Ψ(x,t)Ψ*(x,t)=Exp[−iEh
t]ψ(x)Exp[iEh
t]ψ*(x)
=ψ(x)ψ*(x).They are called stationary states