Agenda for week 1: path integrals

Learning goals:

• Main idea of posing the quantum mechanics axioms in terms of path integrals instead of the usual operator formalism

• Path integrals for a single spinless particle moving in a one-dimensional system.

• Harmonic oscillator propagator and the saddle point/semiclassical approximation

Reading assignment:

Notes for week 1: Path-integral formulation of the problem of one particle on a 1d potential, and the free-particle path integral

More details:

• Tuominen, Chapter 6
• Sakurai: Sec 2.5; Sakurai&Napolitano: Sec 2.6

Preliminary exercises

Do these during/after reading the assignment work. Will be discussed in class March 9th.

1. We have a (single-particle) Hamiltonian , where . Let us find an estimate for the infinitesimal time propagator in the limit .
   1. You can take . Why is this an approximation and why is it justified?
   2. Using this calculate . Use the Gaussian integral result that is given. Do not worry about the convergence of the integrals. In your result identify the Lagrangian.

2. Using the classical action , what is the equation satisfied by the path that minimizes the action? If you use this path to find an approximative result for what are the boundary conditions of this equation?

3. Aharonov-Bohm effect: consider a vector potential
   1. Convince yourself that this corresponds to a magnetic field in the -direction, confined into an infinitesimally small area in the -plane: . Draw a picture with the and field lines.
   2. How do you include the vector potential in the path integral? How does this lead to the Aharonov-Bohm effect?

Homework exercises, first week

Will be discussed in the tutorial session on Thursday March 11th. Return a scanned pdf with your solution by Friday March 12th, at 9 pm using the below form. Then check and grade your solution with the help of the model solutions and resubmit your graded solutions by Monday March 15th at 2 pm.

Prelude: Diffraction with infinitely many slits

Tuominen (Sec 6.1) has a nice justification for the path integral formalism in the form of an "infinite slit diffraction experiment". A brief version of the argument goes as follows:

1. Start from the classical two-slit experiment. A free quantum mechanical particle starts at time , going in the -direction, and passing at time through an infinite, opaque sheet at , and is observed at on a screen behind the sheet. We know that the probability distribution of particles on the screen displays an interference pattern, reflecting the wave-like nature of the particle.
2. How does the interference happen in the formalism? We have a wave function representing the state of the particle. In the case of two slits, the wave function on the screen is a superposition of waves that have passed through the different slits: . We see an interference pattern on the screen, because the probability density reflects the fact that the different slits have different functions .
3. Now consider the case of infinitely many holes in stead of slits. In the same way, our wavefunction is a superposition of infinitely many wavefunctions that have gone through different holes: If the hole lets through particles only at coordinate , we can think of the infinite sum as an integral over the plane But in fact infinitely many holes is the same thing as not having a sheet at all, but just free propagation. Thus this integral just gives the wavefunction of a free particle.
4. In addition to having infinitely many holes, the "no sheet at all" situation can have infinitely many sheets, that the particle passes through at a time , with . Thus our wavefunction of a freely propagating particle an be represented as an infinite dimensional integral

This infinite-dimensional integral is the path integral. It expresses the wavefunction as an integral over all the possible positions of the particle at different times.

The quantity is the wave function (function of ) of a particle that passed through the coordinate at a time . To find what these functions are, we need to look at the time evolution of the wave function, which is given by the Hamiltonian of the system.

Terminology note In classical mechanics, electrodynamics etc. you have encountered things like ; integrals of some function or vector field over a line in 3d space. Such line integrals are sometimes called path integrals. The Feynman path integral is not the same thing. It is an infinite-dimensional integral over paths, i.e. integral over all the possible coordinates that the particle can have at different points in time, not the 1-dimensional integral of some function along a line in 3-dimensional space.

Propagator

Here, we limit the study to the perhaps simplest case of quantum mechanics of one spinless particle in a one-dimensional system. Let us start by reminding of the Schrödinger picture wave function, or the position representation of the state vector, It quantifies the probability amplitude for finding the particle at a certain location at a certain time. In what follows, we concentrate exclusively on the Schrödinger picture and therefore drop the subscript S.

The time evolution of a state in the Schrödinger picture can be written as Then In other words, where the Feynman kernel or the propagator is From above, we see that the "propagation" of the wave function, or the states, from to is controlled by the function .

Especially, if the particle is in an eigenstate of () at the time , the probability amplitude for finding the particle in a state at time is given by the propagator: In the following, let us assume that does not depend on time. Then, as we know, and as a result,

In the exercises, we will compute the propagator of a free particle (). It reads Below we see how propagators can be computed from the path integral.

Decomposition into time intervals

Now, let us develop the path integral formulation for the propagator. We start from a single step: Here we also inserted the resolution of unity . Although the equation as such holds for any set of times, assume here that .

The single propagator thus divides into two: The probability amplitude for finding the particle at a position at a time , when it originally at time was at a location , is given by multiplying the probability amplitudes for the particle to first move from to and then from to , and summing (integrating) over all possible intermediate locations .

This is known as the Markovian property: the probability of finding the particle at depends on its earlier location and the propagator , but not on any memory of what happened before (other than the fact that whatever happened at led to the particle being at ). This property is the basis of the path integral formulation.

Infinitesimal timesteps

Let us now repeat this procedure many times, so that the time interval gets split into intervals of the same length : Thus, , in which and .

The propagator now divides into pieces: Let us now insert a resolution of unity of the form before each of the term of the form , . We get or where , , , and .

Graphically this looks as follows:

In other words, every possible path from to is included!

Explicit form of the Hamiltonian

Now suppose that Then Note that the operators and do not usually commute: . Therefore, we have where the "" indicate higher-order commutators between and . Now, note that when . Thus, we can use For the infinitesimal period we thus have the propagator

From operators to coordinates and momenta

Our infinitesimal time evolution operator is still an operator. The path integral, on the other hand, is expressed in terms of integrals over classical variables. Thus we must get rid of the operators, by introducing complete sets of coordinate of momentum eigenstates. So we start with where we insert

It thus becomes Now the operators operate on their eigenstates so we may replace by and by . Now inserting in front of the first exponential, and in front of the other, we get Now, using the representation of the plane wave states, and the fact that the matrix elements etc., we finally get this to the form Note that this form contains no operators, so the Hamiltonian in the remaining expression is classical.

Substitute this to the equation for the propagator. It becomes Now use so that we get Note that in the limit , the sum in the exponent becomes an integral.

We obtain This is the path integral representation of the propagator. The above formula also defines the sums over all path and momenta, .

Integrating over momenta

In the above expansion, the whole phase space ( and ) is integrated over. We can actually still do the integrals over momenta. For example, where and .

The exponent in the Gaussian integral is

Hence

The Gaussian integral can be found for example from https://en.wikipedia.org/wiki/Gaussian_integral.

Now you should be able to go back and do Question 1 in the preliminary exercises

Configuration space path integral, classical action

The infinitesimal propagator is thus Substituting again this back in Eq. [eq:prop1] gives We hence get the a result for the propagator: We write the right hand side with the notation for the integration measure over all paths, including the normalization. Thus we get the

Feynman path integral representation of the propagator

Here the classical action is and the Lagrange function of the system is

We thus arrived in a configuration () space path integral form of the propagator of the system The integration measure is defined with the above formula.

Interpretation: classical and quantum

From here, we can see how quantum and classical mechanics differ in their behavior with respect to the path integral:

Let us look at the orders of magnitude, considering a classical free particle for which . Then The Euler-Lagrange equation hence yields Then the classical action is Let us put in some numbers corresponding to a macroscopic system, say kg, m/s, s. In this case Now, if is not at its minimum, the weight oscillates rapidly. The contributions from such oscillations thus average out, and only the classical path corresponding to is relevant in macroscopic systems.

For a quantum mechanical system, corresponding to different paths is not large (for example, kg). Thus, in microscopic systems, many paths contribute, yielding quantum effects.

Now you should be able to go back and do Question 2 in the preliminary exercises

In the Feynman path integral formulation of the propagator, Eq. [eq:feynmanpropagator], we can make the following observations:

• The constant , but when computing physical quantities from the path integral, this constant cancels out. This means that the limit should usually be done last.

• Obviously, the whole path integral does not converge even when . We hence need further definitions, which correspond to certain boundary conditions and convergence factors .

Examples of calculable path integrals

Quite generally, you should think of the path integral as an infinitely many dimensional integral, where the integrand is of the form These forms are equivalent, because you can express the time derivative as a difference: thus a function of becomes a function of just at different timesteps, when time is discretized. Such integrals can be computed, when the function is quadratic (i.e. a 2nd degree polynomial) in , i.e. quadratic in .

Particular calculable examples are

• Free particle,

• Harmonic oscillator,

• Forced harmonic oscillator, .

Multidimensional Gaussian integrals

To calculate the calculable path integrals explicitly, we need to take a little and learn how to do Gaussian integrals in many, many dimensions. Given a positive-definite matrix (positive semidefinite meaning here that it is diagonalizable and all the eigenvalues have a positive real part), we want to calculate the -dimensional integral Here is the vector . Note that since is symmetric, a possible antisymmetric part of does not contribute, so we can assume that is symmetric.

Since is real and symmetric, we know (linear algebra truth, not proven here), it can be diagonalized by an orthogonal transformation : where are the eigenvalues of . Then using we get

Now, we want to take the components of the new, rotated vector as our integration variables to perform the Gaussian integral Now we need to change the integration variable from to . The integration measure changes as For this we need the Jacobi determinant of the variable change. To calculate this we calculate the partial derivatives Thus the Jacobi determinant is Because is an orthogonal transformation, the determinant is and thus . We hence get The integrals to be done are simple Gaussian:

We can write this in a form that does not necessarily require diagonalizing the matrix (which could be difficult in the general case), by realizing that it involves the product of the eigenvalues of the matrix, i.e. its determinant: Thus the multidimensional integral is

In principle we could now proeed to straightforwardly evaluate the integral over . However, even if the exponent is purely quardatic in all the coordinates , it is not purely quadratic in the integration variables, because the terms in the sum have terms and that couple the integration variables to the (fixed) boundary conditions. Thus we cannot use the Gaussian integral directly, but would have to generalize it to calculate an integral of the form This can indeed be done. However, let us do this by following an equivalent procedure that has a more transparent physical interpretation, that is discussed in a more general way in the section on the Semiclassical approximation

Instead, we perform a change of integration variables: where is the classical path that satisfies the Euler-Lagrange equation of motion with the boundary conditions . Thus the boundary conditions for the fluctuation around the classical path are (remember that we do not integrate over , but the discretized time derivative couples and to these values).

For this manipulation it is easier to work in the continuum. With the Lagrangian and we have Expanding the squares and integrating over we have three types of terms:

• The classical action
• The cross terms Here we partially integrate and get because the part in the [] brackets is the classical equation of motion.
• The quantum fluctuations around the classical action

Classical action part

We will calculate the classical action part for the harminic oscillator in the exercises. The result is

Quantum fluctuation part

We can again discretize the quantum fluctuation part as Note that this is exactly the same as the original action with replaced by , except that the boundary conditions are now . Also note that does not depend on the coordinates and at all, but only on and , i.e. on the total time interval . Thus the cross term and are not present, and we can directly use the Gaussian integral

In order to write in the form required by the Gaussian integral formula, we need to perform a few manipulations: Now using the fact that we note that the first two sums are equal, and all sums can be written to run from to . We thus get where the latter form takes into account the fact that the product involves a double sum.

We hence obtain from the required symmetric matrix In order for this to be strictly speaking positive definite, some appropriate needs to be introduced, we will not worry about this here.

We will calculate the determinant of this matrix also as a homework exercise. The result is

The result for the path integral now becomes, combining the definition in terms of the path integral, and the Gaussian integral

Finally, the full and exact propagator of the harmonic oscillator is

Point particle path integral in electromagnetic field

Let us add one more ingredient to the path integral description: a classical electromagnetic force. In chapter III we find that the electromagnetic force modifies the Lagrangian as where is the Lagrangian without the electromagnetic field, is the vector potential and is the scalar potential. This thus changes the action to The propagator is thus modified to Including the electromagnetic field thus modifies the phase of each path in a way dependent on the field.

Aharonov-Bohm effect

Now consider describing the two-slit-type experiment in the presence of a magnetic field

Assume there is no scalar potential, i.e., . We wish to find out the effect of the magnetic field on the interference pattern. The propagator in the presence of the background field is now is now where is the action without the gauge potential .

The path integral is a sum over all paths, which must pass through either the slit or slit . The wave function on the screen is just a superposition of two terms which are given (apart from the initial wave function at which is the same for both cases) by Here the restricted measure integrates only over paths that pass through one slit or .

The probability density for finding particles on the screen is . Thus the interference pattern is given by the relative phase of and , i.e. the phase of the product . Let us calculate this product, denoting by the coordinate of the particle on the paths going through slit 1, and by the particles on path 2: We can now change the direction of time on the path y to make it a path starting from on the screen, back to the coordinate of the particle source. This variable change changes the sign of the exponent, and the limits of the path:

Now we can consider this double integration over paths going through slit 1 and paths coming back through slit 2 as closed paths starting from , going though slit 1 to on the screen and then back through slit 2:

But now we recognize a familiar ingredient: for all closed paths C, by Stokes' theorem the integral is just the magnetic flux inside the loop . Thus the wave function product is modulated by a phase factor depending on the magnetic flux between the two sets of paths

This is thus of course nothing but the Aharonov-Bohm effect.

Now you should be able to go back and do Question 3 in the preliminary exercises