1549380323-Statistical Mechanics Theory and Molecular Simulation

380 Quantum mechanics

The probability of obtaining an eigenvalueakofAˆat timetis given by|〈ak|Ψ(t)〉|^2.
Thus, taking the inner product on both sides with〈ak|, we find

〈ak|Ψ(t)〉=〈ak|e−i

Hˆt/ ̄h

|Ψ(0)〉. (9.2.61)

If [A,ˆHˆ] = 0, then|ak〉is an eigenvector ofHˆ with an eigenvalue, sayEk. Hence, the
amplitude for obtainingakat timetis

〈ak|Ψ(t)〉= e−iEkt/ ̄h〈ak|Ψ(0)〉. (9.2.62)

Taking the absolute squares of both sides, the complex exponential disappears, and
we obtain
|〈ak|Ψ(t)〉|^2 =|〈ak|Ψ(0)〉|^2 , (9.2.63)

which implies that the probability at timetis the same as att= 0. Any operator that
is a constant of the motion can be simultaneously diagonalized with theHamiltonian,
and the eigenvalues of the operator can be used to characterize the physical states
along with those of the Hamiltonian. As these eigenvalues are often expressed in terms
of integers, these integers are referred to as thequantum numbersof the state.

9.3 Simple examples

In this section, we will consider two simple examples, the free particleand the harmonic
oscillator, which illustrate how energy quantization arises and how the eigenstates of
the Hamiltonian can be determined and manipulated.

9.3.1 The free particle

The first example is a single free particle in one dimension. In a sense, we solved
this problem in Section 9.2.5 using an argument based on the particle–wave duality.
Here, we will work backwards, assuming eqn. (9.2.44) is true, and solve the eigenvalue
problem explicitly. The Hamiltonian is

Hˆ= ˆp

2

2 m

. (9.3.1)

The eigenvalue problem forHˆcan be expressed as

pˆ^2

2 m

|Ek〉=Ek|Ek〉, (9.3.2)

which, from eqn. (9.2.52), is equivalent to the differential equation

−

̄h^2

2 m

d^2

dx^2

ψk(x) =Ekψk(x). (9.3.3)

Solution of eqn. (9.3.3) requires determining the functionsψk(x), the eigenvaluesEk,
and the appropriate quantum numberk. The problem can be simplified considerably
by noting thatHˆ commutes with ˆp. Therefore,ψk(x) are also eigenfunctions of ˆp,