Peter Atkins • Julio de Paula Atkins’ Physical Chemistry Eighth Edition Chapter 8 Quantum Theory: Introduction and Principles Copyright © 2006 by Peter Atkins and Julio de Paula

Fig 8.29 Integral of the function f(x) = (sin x)(sin 2x) sin x sin 2x

Superpositions and Expectation Values Recall that for a particle in 1-D when A = B, Let’s operate on Ψ with momentum operator, Not an eigenfunction! However, Ψ is a linear combination (sum) of exp(ikx) and exp(-ikx) exp(ikx) and exp(-ikx) correspond to the two momenta Superposition principle: When a system is in a range of states, its Ψ is a linear combination of the Ψi’s: Ψ = c1Ψ1 + c2Ψ2 + ...

Ψ = c1Ψ1 + c2Ψ2 + ... = ΣckΨk Ψk’s are a complete set Quantum mechanically: A single measurement gives the one eigenvalue of that Ψk (that instantaneous state of the system). The probability of a certain eigenvalue being observed is proportional to |ck|2. The average of a large number of observations is given by the expectation value 〈Ω〉 of the operator .                〈Ω〉 is the weighted average of a large number of observations.

Heisenberg uncertainty Fig 8.30 The wavefunction for a 1-D particle at a well-defined location. In this case, Ψ(x) is a wave packet formed by superposition of an infinite number of Ψ(x). What is the particle’s momentum? Quantitatively: In this example, Δq = 0, so Δp = ∞ Heisenberg uncertainty principle

Fig 8.31 The wavefunction for a 1-D particle with an ill-defined location. Examples of complementarity: Position and momentum Wave and particle nature Lifetime and bandwidth

More generally, the uncertainty principle applies to complementary observables in terms of their operators: e.g., If the two observables Ω1 and Ω2 are said to be complementary. When the effect of the operators depends on their order of application (as above), the operators do not commute.

Different outcomes of operator order of application expressed in terms of the commutator: Now, if two operators do not commute, they are complementary, and cannot be determined simultaneously. If two operators commute, they are not complementary, and can be determined with the same degree of precision.

Young’s double-slit experiment Single electrons are emitted from the source at rate of 10/s. Single electron events build up to from an interference pattern in this double-slit experiment.