Ch 9 pages 446-451; 455-463 Lecture 21 – Schrodinger’s equation.

Ch 9pages 446-451; 455-463

Lecture 21 – Schrodinger’s equation

Wave Equations - The Classical Wave Equation

In classical mechanics, wave functions are obtained by solving a differential equation. For example, the one-dimensional wave equation for a vibrating string with linear mass (units of kg/m) and tension (units of force) is:

The wave velocity is given by c and the wave function quantifies the vertical displacement of the string as a function of x and t

0),(),(

2

2

2

2

t

tx

Tx

tx c

T


Any function of the form:

0),(),(

2

2

2

2

t

tx

Tx

tx cT

)/2sin()2cos(),( nnnn xtvAtx

is a solution of the wave equation, where the specific forms for the wave frequency n and the wavelength n are

determined by the details of the problem

For example, for a harmonically vibrating string, fixed at x=0 and x=L (i.e. with boundary conditions 0),(),0( tLt

n n

nc

Land

L

nfor n

2

21 2 3, , ...

The frequencies n are the harmonics of the vibrating string.


Any linear combination of wave functions

0),(),(

2

2

2

2

t

tx

Tx

tx cT


A wave function that is independent of time is called a standing wave. The wave equation for a standing wave is:

n

nn txctx ),(),(

(where cn are constant) is also a solution to the wave equation.

0)()(

2

2

xx

x


Where is a constant. If the boundary conditions are

0),(),(

2

2

2

2

t

tx

Tx

tx cT


then any function of the form:

0)()(

2

2

xx

x

0)()0( L

L

xnx

sin)(

is a solution if 2

L

nn

Standing waves

Wave functions and experimental observables

Particle wave functions are obtained by solving a quantum mechanical wave equation, called the Schroedinger equation

In classical mechanics, the solution to the wave equation (x,t)describes the displacement (e.g. of a string) as a function of time and place

In quantum mechanics, Schrodinger and Heisenberg introduced an analogous concept called wave function (x,t)

Schroedinger’s quantum mechanical wave equation

Schrodinger introduced his famous equation to calculate the value of the wave function for a particle in a potential V(x,t), the time-dependent Schroedinger equation is:

If the potential V is independent of time, the wave function has the simple form:

t

txihtxtxV

x

tx

m

h

),(

2),(),(

),(

8 2

2

2

2

)(),( /2 xCetx hEti

Where C is a constant and satisfies the time-independent Schroedinger’s equation which has a simpler form:

)()(),()(

8 2

2

2

2

xExtxVx

x

m

h

Schroedinger’s quantum mechanical wave equation

Solutions of this equation are independent of time, and are called stationary or standing particle waves, in analogy to classical standing waves in a vibrating string.

t

txihtxtxV

x

tx

m

h

),(

2),(),(

),(

8 2

2

2

2

Schroedinger’s equation for a particle in an infinitely deep well is:

)()(),()(

8 2

2

2

2

xExtxVx

x

m

h

)()(

8 2

2

2

2

xEx

x

m

h

It is identical in form to the classical equation for a standing wave.


Its physical interpretation is not immediate

The square of the wave unction (x,t)2 characterizes the electron distribution in space and is a measure of the probability of finding an electron (or any other particle) at a certain time and place

For example, the probability of finding a particle within a certain volume in space is given by:

V

dxtx ),(2


Although the wave function is a mathematical concept, it is of fundamental importance and can be directly measured (sort of)

For example, X-ray diffraction experiments directly measure (apart from a Fourier transform) the square of the electron distribution of the material

The chemical bond can only be described and understood by calculating wave functions for the electron in a molecule and so on.

Electron density from x-ray crystallography

H E

HThe Hamiltonian: ( )KE PE

The Wavefunction:

E = energy

Describes a system in a given state

The Hamiltonian is an operator

What do we actually measure? Operators

Position x multiply by x

Momentum px

Kinetic energy kx

Potential energy V(x) multiply by V(x)

ihx

2

2m x

Operators are associated with observables


We can calculate the energy of a particle from its wave function and any other property of the system

A fundamental tenet of quantum mechanics is that observables can be derived once the wave function is known.

However, the duality of matter introduced earlier introduces a probabilistic nature to measurements, so that we can calculate observables only in a probabilistic sense


This is done through the expectation value of a variable O, which can be calculated using the expression:

dxOO *

Given a function of a complex variable f=a+ib, the complex conjugate f*=a-ib

For example, the average position of an electron in a molecule is given by:

dxxx *

time-independent Schroedinger equation: particle in a box

Using the Schroedinger Equation we can obtain the energies and wave functions for a particle in a box. Particle-in-a-Box refers to a particle of mass m in a potential defined as:

The wave function has the form

Where (x) is obtained by solving the time-independent Schroedinger equation:

)()()()(

8 2

2

2

2

xExxVx

x

m

h

otherwise

LxifxV

00)(

xetx htEi /2,

With the requirement that 0)()0( L

to reflect the boundary conditions imposed by the potential V(x).


Which can be rearranged to a familiar form

Since the particle must remain in the box where V(x)=0, the Schroedinger equation simplifies to:

The general solution to this equation is:

otherwise

LxifxV

00)( xetx htEi /2,

)(8 2

2

2

2

xExm

h

0)(2

2

2

xx

2

2

2

8

mE

h

xBxAx sincos)(


Can only be satisfied if A=0 and

However, the boundary conditions (the particle cannot leave the box!)

The analogy with standing waves (vibrating strings) is well worth noting. Since, by definition:

otherwise

LxifxV

00)( xetx htEi /2, xBxAx sincos)(

0)()0( L

n

Ln=0, 1, 2, 3, …

2

2

22 8

L

n

h

mE

The quantized energy is obtained by substituting the expression for the wave function into Schrodinger’s equation and solving for the energy.

)(8 2

2

2

2

xExm

h


It is found to be:

We can use the result to calculate the probability that the particle in state n is at position x:

The lowest energy level (n=1) is called ground state, the others are called excited states. These are very important concepts in spectroscopy.

To determine the constants An we can recall that all probabilities

must sum to one because the particle must be somewhere in the box

)(8 2

2

2

2

xExm

h

L

xn

Lx

sin2

)(

En h

mLn 2 2

28

L

xnAxP n

222sin)(

1sin0

22

dx

L

xnA

L

n


Which can be rearranged to give:

Final answer:

L

xnAxP n

222sin)(1sin

0

22

dx

L

xnA

L

n

20

2 1sin

n

L

Adx

L

xn

L

An

22

L

xn

Lx

sin2

)(

• What does the energy look like?

Energy is quantized

E

2 2

28

n hE

mL

n = 1, 2, …

Wavefunctions for particle in the box

• Consider the following dye molecule, the length of which can be considered the length of the “box” an electron is limited to:

2 2

2 2 2 192 2

2 1 2.8 108 8 (8 )final initial

h hE n n x J

mL m Å

What wavelength of light corresponds to E from n=1 to n=2?

L = 8 Å

700nm(experimental: 680 nm)

N

N

+

Application/Example

Solving the Quantum Mechanical Wave Equation

If the potential is independent of time i.e. V=V(x), the Schroedinger equation can be solved as follows.

1. Assume the wave function is a product of a function dependent only on x and a function only dependent on t:

)()(),( txtx

2. Substitute that expression into the Schroedinger equation:

t

tx

ihtxxV

x

tx

m

h

)()(

2)()()(

)()(

8 2

2

2

2

3. Divide both sides of the equation by )()(),( txtx

t

t

t

ihxV

x

x

xm

h

)(

)(

1

2)(

)(

)(

1

8 2

2

2

2

Because the left-hand-side of the equation is dependent only on x, and the right-hand-side is dependent only on t, both sides must equal a constant. It can be shown to be the energy E.


4. The time equation:

This equation has the general solution:

5. The space-dependent equation:

is called the stationary or time independent Schroedinger equation. The solution (x) depends on the potential V(x) and the boundary conditions imposed

Et

t

t

ih

)(

)(

1

2

Can be re-written as: )()(

2tE

t

tih

hEtiet /2)(

)()()()(

8 2

2

2

2

xExxVx

x

m

h


To summarize, the general solution to the Schroedinger equation (if the potential V(x) is independent of time):

is

Where (x) is obtained by solving the time-independent Schroedinger equation:

t

txihtxtxV

x

tx

m

h

),(

2),(),(

),(

8 2

2

2

2

hEtiextx /2)(),(

)()()()(

8 2

2

2

2

xExxVx

x

m

h

Particle in a 3D box

The Schroedinger equation for a particle in a three-dimensional box with dimensions a, b, c is:

This equation can be solved exactly as for a one-dimensional case by assuming:

t

tzyxih

z

tzyx

y

tzyx

x

tzyx

m

h

),,,(

2

),,,(),,,(),,,(

8 2

2

2

2

2

2

2

2

)()()()(),,,( tzyxtzyx zyx

As before hEtiet /2)(

The time-independent wave equation is:

zyx EEEz

tzyx

y

tzyx

x

tzyx

m

h

2

2

2

2

2

2

2

2 ),,,(),,,(),,,(

8 0)()0()()0()()0( cba zzyyxx

Particle in a 3D box

This equation can be further separated into three identical equations of the form of the one-dimensional particle-in-a-box equation. The result is that the energy is a sum of three identical terms:

The wave function is a product of the form:

zyx EEEz

tzyx

y

tzyx

x

tzyx

m

h

2

2

2

2

2

2

2

2 ),,,(),,,(),,,(

8

c

xn

b

xn

a

xn

abczyx zyz

sinsinsin8

),,(

22 2 2

2 2 28yx z

x y z

nh n nE E E E

m a b c

Ch 9 pages 446-451; 455-463 Lecture 21 – Schrodinger’s equation.

Documents

classical wave equation

wave equation x

wave equations

dimensional wave equation

wave velocity

wave function x

classical equation

wave unction x