From wavefunction to state vector

Chapter 2 ended with three questions left hanging: why do observables correspond to operators? What actually happens during a measurement? And what entitles $|c_n|^2$ to be a probability?

Answering them requires switching to a new language. This section explains why the switch is unavoidable.

A clue we have already met

In section 2.10 we described one and the same free particle with two functions:

$$\Psi(x)\quad\text{and}\quad \phi(k)=\frac{1}{\sqrt{2\pi}}\int\Psi(x)\ee^{-\ii kx}\dd x\tag{3.1.1}$$

The two carry exactly the same information: knowing either one lets you compute the other, and normalisation holds on both sides (Parseval’s theorem).

A finite-dimensional analogy

Step back first to a situation you already know well.

Now carry this picture over to quantum mechanics:

$$\underbrace{\Psi(x)}_{\text{components in the position basis}} \qquad \underbrace{\phi(k)}_{\text{components in the momentum basis}} \qquad \underbrace{c_n}_{\text{components in the energy basis}}\tag{3.1.4}$$

Three sets of numbers, one state.

The energy basis: components become a list of numbers

The position and momentum bases are both continuous — the index runs over the whole real line. But bound-state problems come with a more convenient basis.

In section 2.7 we expanded an arbitrary initial state in eigenstates:

$$\Psi(x)=\sum_{n=1}^{\infty}c_n\psi_n(x),\qquad c_n=\int\psi_n^*(x)\Psi(x)\,\dd x\tag{3.1.8}$$

Compare with the vector expansion $\vec v=\sum_i v_i\hat e_i$, $v_i=\hat e_i\cdot\vec v$ — the form is identical.

So in the energy basis, the components of the same state are a list of numbers $(c_1,c_2,c_3,\dots)$.

∎Key formulas

• Components ↔ function values

  $$v_i\ \longleftrightarrow\ \Psi(x)$$

  Read x as a continuous index

• Sum ↔ integral

  $$\sum_i\ \longleftrightarrow\ \int\dd x$$

  Every integral in chapter 2 is an inner product

• Inner product

  $$\vec u\cdot\vec v=\sum_i u_i^*v_i\ \longleftrightarrow\ \int\Phi^*\Psi\,\dd x$$

  Normalisation = length 1

• Three sets of components of one state

  $$\int|\Psi|^2\dd x=\int|\phi|^2\dd k=\sum_n|c_n|^2=1$$

  Length is independent of the choice of basis

?Self-check3 questions

1. 1.

   The most accurate description of the relation between Ψ(x) and φ(k) is:

   • They are two sets of components of the same state vector, in the position and momentum bases
   • Ψ(x) is the real state; φ(k) is merely a mathematical transform
   • They describe two different physical states
   • φ(k) contains more information than Ψ(x)

   Submit
2. 2.

   When a function is viewed as an infinite-dimensional vector, which correspondences are correct? (Select all that apply.)

   Select all that apply

   • Continuous index x ↔ component subscript i
   • ∫dx ↔ Σᵢ
   • ∫Φ*Ψ dx ↔ the inner product Σᵢuᵢ*vᵢ
   • The normalisation condition ↔ the vector having length 1

   Submit
3. 3.

   Why do we say the choice of representation is free?

   • Because the state vector itself is independent of the basis; changing basis only changes the components
   • Because different representations give different physical predictions, so you can pick the right one
   • Because the position representation is always the most convenient
   • Because quantum mechanics allows arbitrary modification of the wavefunction

   Submit

What comes next

If we are going to speak the language of “vectors, bases, components, inner products” for the long haul, we need matching notation — ideally something much shorter to write than $\int\Phi^*(x)\Psi(x)\dd x$.

The notation Dirac designed for the job is one of the most successful notational inventions in all of physics.