Lecture 3

The Schrödinger EquationThe Particle in a Box (part 1)

OrthogonalityPostulates of Quantum Mechanics

NC State University

Chemistry 431

Derivation of the Schrödinger EquationThe Schrödinger equation is a wave equation.

Just as you might imagine the solution of such an equationin free space is a wave. Mathematically we can express awave as a sine or cosine function. These functions are oscillating functions. We will derive the wave equation infree space starting with one of its solutions: sin(x).

Before we begin it is important to realize that bound statesmay provide different solutions of the wave equation thanthose we find for free space. Bound states include rotational and vibrational states as well as atomic wavefunctions. These are important cases that will be treatedonce we have fundamental understanding of the origin ofthe wave equation or Schrödinger equation.

The derivativeThe derivative of a function is the instantaneous rate of change.

The derivative of a function is the slope.

We can demonstrate the derivative graphically. We consider the function f(x) = sin(x) shown below.

The derivative of sin(x)The derivative of a function is the instantaneous rate of change.

The derivative of a function is the slope.

At sin(0) the slope is 1 as shown by the blue line.

The derivative of sin(x)The derivative of a function is the instantaneous rate of change.

The derivative of a function is the slope.

At sin(π/4) the slope is 1/√2 as shown by the blue line.

The derivative of sin(x)The derivative of a function is the instantaneous rate of change.

The derivative of a function is the slope.

At sin(π/2) the slope is 0 as shown by the blue line.

The derivative of sin(x)The derivative of a function is the instantaneous rate of change.

The derivative of a function is the slope.

At sin(3π/4) the slope is -1/√2 as shown by the blue line.

The derivative of sin(x)The derivative of a function is the instantaneous rate of change.

The derivative of a function is the slope.

At sin(3π/4) the slope is -1/√2 as shown by the blue line. The slopes of all lines thus far are plotted as black squares.

The derivative of sin(x)The derivative of a function is the instantaneous rate of change.

The derivative of a function is the slope.

At sin(π) the slope is -1 as shown by the blue line.The slopes of all lines thus far are plotted as black squares.

The derivative of sin(x)The derivative of a function is the instantaneous rate of change.

The derivative of a function is the slope.

At sin(5π/4) the slope is -1/√2 as shown by the blue line.The slopes of all lines thus far are plotted as black squares.

The derivative of sin(x)The derivative of a function is the instantaneous rate of change.

The derivative of a function is the slope.

At sin(5π/4) the slope is -1/√2 as shown by the blue line.The slopes of all lines thus far are plotted as black squares.

The derivative of sin(x)The derivative of a function is the instantaneous rate of change.

The derivative of a function is the slope.

We see from of the black squares (slopes) that the derivative of sin(x) is cos(x).

The derivative of sin(x)

sin(x) = cos(x)ddx

The derivative of cos(x)

cos(x) = -sin(x)ddx

The second derivative of sin(x)

sin(x) = -sin(x)ddx

ddx

The second derivative of sin(x)

sin(x) = -sin(x)d2

dx2

Sin(x) is an eigenfunction

sin(x) = sin(x)d2

dx2

If we define as an operator G then we have:

-

d2

dx2-

G sin(x) = sin(x)

which can be written as:

This is a simple example of an operator equation thatis closely related to the Schrödinger equation.

Sin(kx) is also an eigenfunction

sin(kx) = -k cos(kx)ddx

We can make the problem more general by includinga constant k. This constant is called a wavevector.It determines the period of the sin function. Now we musttake the derivative of the sin function and also the function kx inside the parentheses (chain rule).

-

Here we call the value k2 the eigenvalue.

sin(kx) = k2sin(kx)d2

dx2-

Sin(kx) is an eigenfunction of the Schrödinger equation

The example we are using here can easily be expressedas the Schrodinger equation for wave in space. We onlyhave to add a constant.

In this equation h is Planck’s constant divided by 2π andm is the mass of the particle that is traveling through space.The eigenfunction is still sin(kx), but the eigenvalue in this equation is actually the energy.

sin(kx) = sin(kx)d2

dx2-

h2

2m

- h2k2

2m

-

-

The Schrödinger equationBased on these considerations we can write a compactform for the Schrödinger equation.

HΨ = EΨ

d2

dx2-

h2

2m

-

E =h2k2

2m

-

H =

Ψ= sin(kx)

Energy operator, Hamiltonian

Energy eigenvalue, Energy

Wavefunction

The momentumThe momentum is related to the kinetic energy. ClassicallyThe kinetic energy is:

E = mv2

The momentum is:p = mv

So the classical relationship is:p2

E = 2m

If we compare this to the quantum mechanical energy:

we see that: p = hk

1

E =h2k2

2m

-

2

The general solution to the Schrödinger equation in free space

The preceding considerations are true in free space.Since a cosine function has the same form as a sinefunction, but is shifted in phase, the general solutionis a linear combination of cosine and sine functions.

The coefficients A and B are arbitrary in free space.

However, if the wave equation is solved in the presenceof a potential then there will be boundary conditions.

Ψ= Asin(kx) + Bcos(kx) Wavefunction

The particle in a box problemImagine that a particle is confined to a region of space.The only motion possible is translation. The particle hasonly kinetic energy. While this problem seems artificial atfirst glance it works very well to describe translational motion in quantum mechanics.

0 LAllowed Region

The solution to the Schrödinger equation with boundary conditions

Suppose a particle is confined to a space of length L.On either side there is a potential that is infinitely large.The particle has zero probability of being found at the boundary or outside the boundary.

0 LAllowed Region

The solution to the Schrödinger equation with boundary conditions

The boundary condition is that the wave function willbe zero at x = 0 and at x = L.

From this condition we see that B must be zero.This condition does not specify A or k.The second condition is:

From this condition we see that kL = nπ. The conditionsso far do not say anything about A. Thus, the solutionfor the bound state is:

Note that n is a quantum number!

Ψ(0) = Asin(k0) + Bcos(k0) = 0

Ψ(L) = Asin(kL) = 0 or kL = arcsin(0)

Ψn(x) = Asin(nπx/L)

The probability interpretationThe wave function is related to the probability for findinga particle in a given region of space. The relationship isgiven by:

If we integrate the square of the wave function over a given volume we find the probability that the particle isin that volume. In order for this to be true the integralover all space must be one.

If this equation holds then we say that the wave functionis normalized.

P = Ψ2dV

1 = Ψ2dVall space

The normalized bound state wave function

For the wave function we have been considering, all space is from 0 to L. So the normalization constant Acan be determined from the integral:

The solution to the integral is available on the downloadable MAPLE worksheet. The solution is just L/2.Thus, we have:

As you can see the so-called normalization constant hasbeen determined.

1 = Ψ2dx0

L

= A2 sin nπxL

2dx

0

L

= A2 sin nπxL

2dx

0

L

1 = A2 L2 , A2= 2

L , A = 2L

NormalizationWhat is the normalization constant for the wave functionexp(-ax) over the range from 0 to infinity?

A. aB. 2aC. √aD. √2a

NormalizationWhat is the normalization constant for the wave functionexp(-ax) over the range from 0 to infinity?

A. aB. 2aC. √aD. √2a

1 = Ψ2dx0

L

= A2exp – ax 2dx0

∞

= A2 exp – 2ax dx0

∞

= A2 12a , A = 2a

The appearance of the wave functions

The appearance of the wave functions

Note that the wave functions havenodes (i.e. the locations where theycross zero). The number of nodes isn-1 where n is the quantum numberfor the wave function. The appearanceof nodes is a general feature of solutions of the wave equation in bound states. By boundstates we mean states that are in a potential such as theparticle trapped in a box with infinite potential walls. Wewill see nodes in the vibrational and rotational wave functionsand in the solutions to the hydrogen atom (and all atoms). Note that the wave functions are orthogonal to one another.This means that the integrated product of any two of these functions is zero.

The energy levelsThe energy levels are:

E = n2h2

8mL2

The probability of finding theparticle in a given region of space

Using the normalized wave function

one can calculate the probability of finding the particle in any region of space. Since the wave function is normalized, the probability P is a number between 0 and 1.For example: What is the probability that the particle isbetween 0.2L and 0.4L. This is found by integrating overthis region using the normalized wave function (see MAPLE worksheet).

Ψ x = 2L sin nπx

L

P = Ψ x 2dx0.2L

0.4L

= 2L sin nπx

L2dx

0.2L

0.4L

≈ 0.25

The appearance of the probability Ψ2

ö

The probability of finding theparticle in a given region of space

Using the normalized wave function

one can calculate the probability of finding the particle in any region of space. Since the wave function is normalized, the probability P is a number between 0 and 1.For example: What is the probability that the particle isbetween 0.2L and 0.4L. This is found by integrating overthis region using the normalized wave function (see MAPLE worksheet).

Ψ x = 2L sin nπx

L

P = Ψ x 2dx0.2L

0.4L

= 2L sin nπx

L2dx

0.2L

0.4L

≈ 0.25

Solutions of the Schrodinger equationare orthogonal

If the wavefunctions have different quantum numbersThen their “overlap” is zero. We can call the integralOf the product of two wavefunctions an overlap. We write:

Where δmn is called the Kronecker delta. It has the property:

For the particle-in-a-box the orthogonality is written:

2L sin mπx

L sin nπxL dx = δmn

0

L

ΨmΨndx0

L

= δmn

δmn =0 if m ≠ n1 if m = n

Postulates of quantum mechanics are assumptions found to be consistent with observation

The first postulate states that the state of a system can be represented by a wavefunction Ψ(q1, q2,.. q3n, t). The qi are coordinates of the particles in the system and t is time.

The wavefunction can also be time-independent or stationary, ψ(q1, q2,.. q3n).

Corollary: An acceptable wavefunction must be continuous and

have a continuous first derivative

Since the wavefunction is a solution of the Schrödinger equation it must be differentiable.

The wavefunction must also be single-valued.

Postulate 2. The probability of finding a particle in a region of

space is given by

P(a) = Ψ∗Ψdτ0

a

Assumptions for the Born interpretation1. Ψ*Ψ is real (Ψ is Hermitian).2. The wavefunction is normalized.3. We integrate over all relevant space.

Normalization is needed so that probabilities are meaningful

Ψ∗Ψdτall space

= 1

Normalization means that the integral of thesquare of the wavefunction (probability density)over all space is equal to one.

The significance of this equation is that theprobability of finding the particle somewherein the universe is one.

Postulate 3. Every physical observable is associated with a

linear Hermitian operator

operator P → observable P

Observables are energy, momentum, position,dipole moment, etc.

The fact that the operator is Hermitian ensuresthat the observable will be real.

Postulate 4. The average value of a physical property

can be calculated by

P =Ψ∗PΨdτ

Ψ∗Ψdτ

Normalization

Postulate 4. The calculation of a physical observable can be

written as an eigenvalue equation

PΨ = PΨThis is an operator equation that returns thesame wavefunction multiplied by the constant P.P is an eigenvalue. An eigenvalue is a number.

The form of the operators is

Position q q

Momentum P ih ∂∂q

Time t t

Energy H ih ∂∂t

Postulate 5. It is impossible to specify with arbitrary precision both the

position and momentum of a particle

This postulate is known as the HeisenbergUncertainty Principle. It applies not only to the pair of variables position and momentum,but also to energy and time or any two conjugatevariables. Conjugate variables are Fourier transforms of one another.Conjugate variables do not commute.

Commutator in quantum mechanicsA commutator is an operation that compares the order of operation for two operators:

Since the momentum, p involves a derivative, the order of application affects the result. The way to see the result of the commutator is to apply it to a test function f(x). pxf(x) = -ih(f(x) + xf’(x)) while xpf(x)= -ihxf’(x). Therefore [p,x] = - ih.We say that position and momentum do not Commute.

[ p,x] = px – x p