lecture 3: the time dependent schrödinger equation

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


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


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)


−

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


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


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




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


Time independent Schrödinger equation


−

h2

2mδ2ψ(x)

δx2 +ψ(x)V(x)=Eψ(x)

Or:




−

h2

2mδ2ψ(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)




If ψ( )x is a solution to

−

h2

2mδ2ψ(x)


So is Aψ( )x

−

h2

2mδ2(Aψ(x))

δx2 +Aψ(x)V(x)=AEψ(x)


Lecture 2



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)


time independent probability function


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)




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.




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


−

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

lecture 3: the time dependent schrödinger equation

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