1.2: Recap- Position and Momentum States
( \newcommand{\kernel}{\mathrm{null}\,}\)
Before proceeding, let us review the properties of quantum particles in free space. In a d-dimensional space, a coordinate vector r is a real vector of d components. A quantum particle can be described by the position basis—a set of quantum states {|r⟩}, one for each possible r. If we are studying a particle trapped in a finite region (e.g., a particle in a box), r is restricted to that region; otherwise, r is any real d-dimensional vector. In either case, the r’s are continuous, so the position eigenstates form an uncountably infinite set.
The position eigenstates are assumed to span the state space, so the identity operator can be resolved as
ˆI=∫ddr|r⟩⟨r|,
where the integral is taken over all allowed r. It follows that
⟨r|r′⟩=δd(r−r′).
The position eigenstates are thus said to be “delta-function normalized”, rather than being normalized to unity. In the above equation, δd(⋯) denotes the d-dimensional delta function; for example, in 2D,
⟨x,y|x′,y′⟩=δ(x−x′)δ(y−y′).
The position operator ˆr is defined by taking |r⟩ and r as its eigenstates and eigenvalues:
ˆr|r⟩=r|r⟩.
Momentum eigenstates are constructed from position eigenstates via Fourier transforms. First, suppose the allowed region of space is a box of length L on each side, with periodic boundary conditions in every direction. Define the set of wave-vectors k corresponding to plane waves satisfying the periodic boundary conditions at the box boundaries:
{k|kj=2πm/Lform∈Z,j=1,…,d}.
So long as L is finite, the k vectors are discrete. Now define
|k⟩=1Ld/2∫ddreik⋅r|r⟩,
where the integral is taken over the box. These can be shown to satisfy
⟨k|k′⟩=δk,k′,⟨r|k′⟩=1Ld/2eik⋅r,I=∑k|k⟩⟨k|.
The momentum operator is defined so that its eigenstates are {|k⟩}, with ℏk as the corresponding eigenvalues:
ˆp|k⟩=ℏk|k⟩.
Thus, for finite L, the momentum eigenstates are discrete and normalizable to unity. The momentum component in each direction is quantized to a multiple of Δp=2πℏ/L.
We then take the limit of an infinite box, L→∞. In this limit, Δp→0, so the momentum eigenvalues coalesce into a continuum. It is convenient to re-normalize the momentum eigenstates by taking
|k⟩→(L2π)d/2|k⟩.
In the L→∞ limit, the re-normalized momentum eigenstates satisfy
Definition: Re-normalized momentum eigenstates
|k⟩=1(2π)d/2∫ddreik⋅r|r⟩,|r⟩=1(2π)d/2∫ddke−ik⋅r|k⟩,⟨k|k′⟩=δd(k−k′),⟨r|k⟩=1(2π)d/2eik⋅r,I=∫ddk|k⟩⟨k|.
The above integrals are taken over infinite space, and the position and momentum eigenstates are now on a similar footing: both are delta-function normalized. In deriving the above equations, it is helpful to use the formula
∫∞−∞dxexp(ikx)=2πδ(k).
For an arbitrary quantum state |ψ⟩, a wavefunction is defined as the projection onto the position basis: ψ(r)=⟨r|ψ⟩. Using the momentum eigenstates, we can show that
⟨r|ˆp|ψ⟩=∫ddk⟨r|k⟩ℏk⟨k|ψ⟩=∫ddk(2π)d/2ℏkeik⋅r⟨k|ψ⟩=−iℏ∇∫ddk(2π)d/2eik⋅r⟨k|ψ⟩=−iℏ∇ψ(r).
This result can also be used to prove Heisenberg’s commutation relation [ˆri,ˆpj]=iℏδij.