Physical Chemistry Third Edition

712 16 The Principles of Quantum Mechanics. II. The Postulates of Quantum Mechanics

The productσxσpxis a measure of the combined uncertainty ofxandpx, and is called

anuncertainty product. This kind of uncertainty product is involved in the Heisenberg

uncertainty principle. From Examples 16.16 and 16.21 the value of the uncertainty

product ofxandpxfor then1 state of the particle in a one-dimensional box is

σxσpx( 0. 180756 a) ̄

hπ

a

 0. 56786 h ̄ 0. 09038 h (16.5-1)

Table 16.1 gives some values ofσx,σpx, andσxσpxfor several states of a particle in a

one-dimensional box.

The coordinatexand the momentum componentpxare examples ofconjugate

variablesin the sense of Eq. (E-20) of Appendix E. The operators of this pair of variables

do not commute, as shown in Example 16.4. TheHeisenberg uncertainty principleis

a general statement of the combined uncertainties of two conjugate variables:The

product of the uncertainties of two conjugate variables is always equal to or larger

thanh/ 4 π, wherehis Planck’s constant.If we use the symbols∆xand∆pxfor

the uncertainties of a coordinate and its conjugate momentum, then the uncertainty

principle is

∆x∆pxε≥

h

4 π

 ̄

h

2

(uncertainty principle) (16.5-2)

The minimum value in Eq. (16.5-2) corresponds to the use of standard deviations as

measures of uncertainty:

∆xσx (16.5-3)

and

∆pxσpx (16.5-4)

Table 16.1 Values of the Uncertainty Product for a Particle

in a One-Dimensional Box

n σx σpx σxσpx

1 0.18076ah/2a 0.09038h0.56786h ̄

2 0.26583ah/a 0.26583h1.67029h ̄

3 0.27876a 3 h/2a 0.41813h2.62720h ̄

... ... ... ...

∞ 0.28868a ∞∞

Numerical Values for a Box of Length 10.0× 10 −^10 m

(Model for Pi Electrons in 1,3,5-Hexatriene)

nσx/m σpx/kg m s−^1 σxσpx/kg m^2 s−^1

11. 808 × 10 −^103. 313 × 10 −^255. 909 × 10 −^35

22. 626 × 10 −^106. 626 × 10 −^251. 761 × 10 −^34

32. 788 × 10 −^109. 939 × 10 −^252. 771 × 10 −^34

... ... ... ...

∞ 2. 887 × 10 −^10 ∞∞