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Wave Functions in X-P Space

In this note, we will discuss the wavefunctions in the position-momentum space. We will see how the wave functions in the position space and the momentum space are related to each other.

Table of Contents​

The Wavefunction​

Recall that the position kets are the eigenkets of the position operator :

The eigenkets are orthogonal;

We have briefly discussed the wavefunction in the first section of this chapter. They are the continuous analog of the coefficients of the state vector in a discrete basis:

We define the coefficient as the wavefunction :

Wavefunction in the Position Space:

For an inner product of two states and , we have:

If we take an inner product of a state vector with an eigenket , we get the following:

The term is the wavefunction in the basis. The term is the overlap between the position ket and the basis ket . It is known as an eigenfunction of the operator with the eigenvalue :

Next, consider the expression , where is an operator. We can write this as:

The central term is the matrix element of the operator in the position basis. If is a function of the position operator , i.e. , then the matrix element is given by:

Then the expression becomes:

Momentum Operator on Position Eigenkets​

Recall that the translation operator translates the position ket by and, in the infinitesimal case, is given by:

Applying it to a state vector and expanding in the position basis, we get:

where we have shifted the variable of integration in the last step. The term is equal to , where we have used the Taylor expansion of the wavefunction around :

Using the expression for the translation operator , we can write the above equation as:

Expanding the left-hand side of the equation, we get:

Recall that , so we can write the above equation as:

Now move the integral from the left-hand side to the right-hand side:

Both integrals have a term, so they cancel out and we are left with:

Canceling the term and rearranging and on the left-hand side, we get:

Taking the inner product of the Equation with a position eigenket gives:

Taking the inner product of the Equation with another ket gives:

The above is usually what introductory textbooks present at the beginning of the chapter on quantum mechanics. In our formalism, we have derived the above equation from properties such as the translation operator and the canonical commutation relation.

To repeatedly apply the momentum operator to a state vector, we can use the following recursive formula:

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It is interesting to see the parallel with generators once again. Recall that momentum is the generator of translations in the position space. In classical mechanics, we can write this in terms of the Lagrangian as:

The parallel with the quantum mechanical expression is striking:

Wavefunction in the Momentum Space​

So far we have discussed the wavefunction in the position space. In the momentum space, the wavefunction is defined as:

where are the eigenkets of the momentum operator :

Similar to the position eigenkets, the momentum eigenkets are orthogonal:

And they span the space:

Transformation between Position and Momentum Space​

Recall that we change basis by using a certain matrix. From an old set of bases to a new set of bases , we have:

To derive the transformation between the position and momentum space, we will use the position and momentum operators as the basis kets. First, the inner product can be calculated by using Equation and setting :

Since , we can write the above equation as:

This is a simple first-order differential equation. If not immediately obvious, it has a form equivalent to , where is a constant. The solution to this differential equation is . In our case, and . Thus, the solution is:

The key insight comes from the following: consider a position wavefunction . We can expand this wavefunction in the momentum basis:

Now that we know the form of , we can write the above equation as:

This means that the wavefunction is a superposition of plane waves—a Fourier series!

To find out what is, we can use the orthogonality of the eigenkets. First consider the inner product :

The left-hand side is . The right-hand side can be calculated by using the expression for and :

The integral is a Dirac delta function because it satisfies the defining property . Thus, the integral evaluates to . Rearranging the terms, we get:

Thus, the wavefunction in the momentum space is:

A similar derivation can be done for the wavefunction in the position space:

It is important to note that the wavefunctions in the position and momentum space are Fourier transforms of each other. See this page for a refresher on Fourier series.

Application: Gaussian Wavefunction​

A Gaussian wave packet is a wavefunction that is localized in both the position and momentum space. It is given by:

This is known as a Gaussian wave packet because its probability density is a Gaussian function (a bell curve). It looks like this:

The probability amplitude, , is a Gaussian function in the position space. It looks like this:

As you can see, it quickly decays as we move away from the center of the wave packet.

Expectation Value of Position​

The expectation value of the position operator is given by:

This is because the wave packet is symmetric around the origin.

Uncertainty in Position​

To calculate the dispersion in position, we use:

(This comes from the definition of the variance of a random variable.)

The expectation value of the position squared operator is given by:

where we have used the Gaussian integral .

Then, the dispersion in position is:

Expectation Value of Momentum​

The expectation value of the momentum operator is given by:

The first integral is a standard Gaussian integral and results in . The second integral is zero because the integrand is an odd function. Thus, the expectation value of the momentum is:

Uncertainty in Momentum​

We first calculate the expectation value of the momentum squared operator :

where we have used the Gaussian integrals and .

Then, the dispersion in momentum is:

Heisenberg Uncertainty Principle​

Now that we have calculated the dispersions in position and momentum, we can calculate the product of the dispersions:

In this case, the product of the dispersions is a constant, which is the minimum value allowed by the Heisenberg Uncertainty Principle.

Momentum-Space Wavefunction​

The wavefunction in the momentum space can be calculated by using the Fourier transform of the position-space wavefunction:

We can complete the square in the exponent——to get:

Thus:

Making the substitution , the integral becomes , which is a Gaussian integral and evaluates to .

Thus, the wavefunction in the momentum space is:

Both the position and momentum space wavefunctions are Gaussian functions and are shown below:

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Notice that (when you move the slider for ) the wave packet in the position space becomes narrower as the wave packet in the momentum space becomes wider, and vice versa. This is a manifestation of the uncertainty principle which, in this case, according to Equation , is .

In the limit , the wave packet in the position space becomes a plane wave and the wave packet in the momentum space becomes a delta function. Probabilistically, this means that the position of the particle is completely unknown, but its momentum is known exactly.

Conversely, in the limit , the wave packet in the position space becomes a delta function and the wave packet in the momentum space becomes a plane wave. This means that the position of the particle is known exactly, but its momentum is completely unknown.

Summary and Next Steps​

In this chapter, we have introduced the wavefunction in the position and momentum space. We have derived the transformation between the two spaces and calculated the wavefunction of a Gaussian wave packet in both spaces.

Below are the important equations derived in this chapter, generalized to three dimensions:

  • Position eigenket:

  • Momentum eigenket:

  • Three-dimensional Dirac delta function:

  • Orthogonality of eigenkets:

  • Completeness relation:

  • Spectral decomposition of state vector:

  • Momentum operator on position-space wavefunctions:

  • Coordinate transformation:

  • Fourier transforms of position and momentum space wavefunctions:

This completes the first part of our journey into quantum mechanics—understanding the mathematical foundations of the state vector formalism. The next page is a summary of all important results derived in this chapter.