Lecture 4Partial differentiation

(c) So Hirata, Department of Chemistry, University of Illinois at Urbana-Champaign. This material has been developed and made available online by work supported jointly by University of Illinois, the

National Science Foundation under Grant CHE-1118616 (CAREER), and the Camille & Henry Dreyfus Foundation, Inc. through the Camille Dreyfus Teacher-Scholar program. Any opinions, findings, and conclusions or recommendations expressed in this material are those of the author(s) and do not

necessarily reflect the views of the sponsoring agencies.

Partial differentiation

In quantum chemistry, we often deal with a function of more than just one variables, for example, f(x,y,z).

When considering the derivatives of a multi-variable function, we must be aware of with respect to which variable we are differentiating the function.

is used to indicate the derivative is with respect to z, and x and y are held fixed in the differentiation.yxz

f

,

Partial differentiation

∂ (round d) indicates the partial differentiation.

Consider a function of space (x) and time (t) variables, f(x, t). Let the space variable also depend on time x = x(t). In this case, a partial derivative of f with respect to t is different from the exact derivative because

xtx t

f

dt

dx

x

f

t

f

dt

txdf

),(

Partial differentiation

Other than that, partial differentiation follows essentially the same rules as usual differentiation.

Is this true?

YES and NO - this is only true if the variables held fixed are identical in the left- and right- hand sides.

1y x

x y

Partial differentiation

Consider the change in function f(x,y,z) caused by an increase in x (y and z held fixed) and then in y (x and z held fixed).

The result would be the same if we increase y first and then x.

xy

f

yx

f

22

Partial differentiation

xy

f

yx

f

22

Partial differentiation

22

3axyx

f

23),( byyaxyxf

yaxx

f

y

23

byaxy

f

x

23

22

3axxy

f

Time-dependent Schrödinger equation We use partial derivatives for operators. For example, the energy operator is,

We do not differentiate x, y, z dependent part of the wave function by t (see the simple wave in the previous lecture)

zyxtiE

,,

Time-dependent Schrödinger equation

The time-dependent Schrödinger equation is:

zyxtiH

,,

ˆ

The Schrödinger equation

For one-dimension, it is

The kinetic energy operator comes from the classical to quantum conversion of the momentum operator

dx

dip

The Schrödinger equation In three-dimension, we have three Cartesian

components of a momentum:

Accordingly, the momentum operator is a vector operator:

(“del”) is a vector

zip

yip

xip zyx

;;

iz

iy

ix

ipppp zyx ,,),,(

zyx /,/,/

The Schrödinger equation

Kinetic energy in classical mechanics:

(The vector inner product is )

In quantum mechanics:

m

ppp

m

p zyx

22

2222

222zyx ppppp

2

2

2

2

2

222

2

22 zyxmm

The Schrödinger equation

The Schrödinger equation in three dimensions is,

The Schrödinger equation in spherical coordinates

Instead of Cartesian coordinates (x, y, z), it is sometimes more convenient to use spherical coordinates (r, θ, φ)

cos

sinsin

cossin

rz

ry

rx

The Schrödinger equation

EzyxV

mH ),,(

22

2

2

2

2

2

2

22

zyx

sin

sin

1

sin

1122

2

222

22

rrrr

The kinetic energy operator can be written in two ways – Cartesian or spherical.

Homework Challenge #2

Derive the spherical-coordinate expression of (the green panel) using the equations in blue and orange panels.

2

2

2

2

2

22

zyx

sin

sin

1

sin

1122

2

222

22

rrrr

cos

sinsin

cossin

rz

ry

rx

2

Summary

We use partial derivatives to define quantum-mechanical operators.

In using partial derivatives, we must be aware of which variables are being held fixed.