Quantum One: Lecture 11

The Position and the Momentum Representation

In the last lecture, we defined for a single quantum particle, the position representation generated by the ONB of position states | , in terms of which a ⟩state |ψ can be represented by a wave function ⟩ ψ() = |⟨ ψ , and inner products ⟩between different states can be computed via the relation

⟨χ|ψ⟩ = ∫ d³r ) ψ()

We then used the orthonormal set of momentum eigenfunctions to define a momentum or wavevector representation generated by states | , defined in ⟩terms of the basis states of the position representation through the relation

and in terms of which the states of the position representation can be written

In this lecture we extend our understanding of the relationship between the position and the momentum representation.

To this end, we note that that the momentum states | , which form an ONB for ⟩the space, generate a representation for it. This means that an arbitrary state |ψ ⟩of this space can be expanded in this set of states in terms of a momentum space wave function ψ(), so that

where

in terms of which we can compute the inner product between two arbitrary states through the relation

⟨χ|ψ⟩ = ∫ d³k ) ψ()

just as we did in the position representation.

In this lecture we extend our understanding of the relationship between the position and the momentum representation.

To this end, we note that that the momentum states | , which form an ONB for ⟩the space, generate a representation for it. This means that an arbitrary state |ψ ⟩of this space can be expanded in this set of states in terms of a momentum space wave function ψ(), so that

where

in terms of which we can compute the inner product between two arbitrary states through the relation

⟨χ|ψ⟩ = ∫ d³k ) ψ()

just as we did in the position representation.

In this lecture we extend our understanding of the relationship between the position and the momentum representation.

To this end, we note that that the momentum states | , which form an ONB for ⟩the space, generate a representation for it. This means that an arbitrary state |ψ ⟩of this space can be expanded in this set of states in terms of a momentum space wave function ψ(), so that

where

and in terms of which we can compute the inner product between two arbitrary states through the relation

⟨χ|ψ⟩ = ∫ d³k ) ψ()

just as we did in the position representation.

This expansion of an arbitrary state in the momentum representation, also allows us to figure out how to transform from the position representation to the momentum representation, and vice versa, i.e. we note that since

then

which shows that ψ() is the Fourier transform of ψ().

This expansion of an arbitrary state in the momentum relation, also allows us to figure out how to transform from the position representation to the momentum representation, and vice versa, i.e. we note that since

then

which shows that ψ() is the Fourier transform of ψ().

This expansion of an arbitrary state in the momentum relation, also allows us to figure out how to transform from the position representation to the momentum representation, and vice versa, i.e. we note that since

then

which shows that ψ() is the Fourier transform of ψ().

This expansion of an arbitrary state in the momentum relation, also allows us to figure out how to transform from the position representation to the momentum representation, and vice versa, i.e. we note that since

then

which shows that ψ() is the Fourier transform of ψ().

This expansion of an arbitrary state in the momentum relation, also allows us to figure out how to transform from the position representation to the momentum representation, and vice versa, i.e. we note that since

then

which shows that ψ() is the Fourier transform of ψ().

Similarly, we find that, since

then

so that

and

are Fourier transform pairs. Note and know where the plus and minus signs go!

Similarly, we find that since

then

so that

and

are Fourier transform pairs. Note and know where the plus and minus signs go!

Similarly, we find that since

then

so that

and

are Fourier transform pairs. Note and know where the plus and minus signs go!

Similarly, we find that since

then

so that

and

are Fourier transform pairs. Note and know where the plus and minus signs go!

Similarly, we find that since

then

so that

and

are Fourier transform pairs. Note and know where the plus and minus signs go!

Similarly, we find that since

then

so that

and

are Fourier transform pairs. Note and know where the plus and minus signs go!

Similarly, we find that since

then

so that

and

are Fourier transform pairs. Note and know where the plus and minus signs go!

Similarly, we find that since

then

so that

and

are Fourier transform pairs. You should note carefully where the plus and minus signs go!

Also note a subtle shift in notation.

Previously we denoted the wave function by ψ() and its Fourier transform by ().

We now know, however, that there are as many possible wave functions representing the state |ψ as there are continuous ONB’s for the space. ⟩

So, rather than coming up with a different accent mark ψ, , , etc. ,we agree to always just include the argument of the wave function to indicate which representation we are working in at the moment.

Thus, it will be understood that

ψi = |⟨ ψ ⟩ ψ () = |⟨ ψ ⟩ ψ() = |⟨ ψ ⟩ ψ(α) = |⟨ ψ ⟩

all represent different functions of their index, or argument, even though we use the same symbol ψ for each, which refers to the same underlying state vector |ψ .⟩

In finishing up our discussion of the first postulate, it is useful to explicitly gather together the important relations regarding the position and momentum representation, which allows the symmetry between them to be appreciated.

Continued:

We often work on problems that involve motion only in one dimension, for which the following analogous relations hold:

We have completed our discussion of the first postulate of the general formulation of quantum mechanics, and illustrated its structure by using our general formal definitions about state vectors in quantum mechanical state space, to re-derive properties that were familiar to use from Schrödinger's mechanics for a single quantum particle.

In the next lecture, we begin a discussion of the second postulate, which provides information about the nature of observables of general quantum mechanical systems.

We have completed our discussion of the first postulate of the general formulation of quantum mechanics, and illustrated its structure by using our general formal definitions about state vectors in quantum mechanical state space, to re-derive properties that were familiar to use from Schrödinger's mechanics for a single quantum particle.

In the next lecture, we begin a discussion of the second postulate, which provides information about the nature of observables of general quantum mechanical systems.