Problem: There is a quantum, non-relativistic free particle in a one-dimensional space. Describe a method to determine its wave function ψ(x) or density operator ρ using a harmonic oscillator potential that can be applied in an experiment and a measuring device that measures the position of the particle.

It is a regrettable reality that the method for uniquely determining a particle's wave function ψ(x) or density operator ρ from experimental data of probability distributions of physical quantities has not yet permeated many physics researchers in various fields. Some professors are even surprised when told that "the wave function is determined precisely from experiments alone, excluding the overall phase factor that does not affect physics." When asked again about their definition of a wave function, the overwhelming majority of such people answer simply that it is "a complex function satisfying the Schrödinger equation, from which the probability distribution of physical quantities can be calculated using the Born rule for that function." However, this is merely the content of old quantum mechanics textbooks from the last century. Or, there are quite a few physics professors with a mathematical bent who answer that "a wave function is an element of a Hilbert space, introducing equivalence classes of functions that match almost everywhere."

However, even if one can calculate the predictions for experiments provided by the wave function, this does not lead to a physical understanding of the wave function itself. The wave function is uniquely defined from a set of probability distributions of specific physical quantities measured in experiments. In other words, the framework of "wave function = probability distribution" is clear today, but there are still many professors who have not updated their knowledge. Please also see the article below regarding this current situation.

With the aim of improving this situation, I published a book titled "New Common Sense of Quantum Mechanics" together with Katsuhiko Hiroe.

This is also positioned as a supplementary reader for "Introduction to Modern Quantum Mechanics," which is commonly referred to as "Hotta Quantum Mechanics."

This "New Common Sense of Quantum Mechanics" contains a wealth of content not only on how to define wave functions based on experiments, but also on resolving various mistakes and misunderstandings written in physics textbooks of the last century. It is a book that I would like not only physics students learning quantum mechanics to read, but also many people who have previously studied quantum mechanics at university.

Now, in the following, I will introduce the "quantum state tomography method" for defining the wave function or density operator of a particle in a one-dimensional space. This is content written in Appendix B of "New Common Sense of Quantum Mechanics" and Appendix A of "Quantum Information and the Physics of Spacetime," which was published before that.

As a method for determining the density operator ρ at t=0 for a particle of mass m undergoing free motion without a potential at times t<0, we consider a quantum state tomography method that uses a device to apply a harmonic oscillator potential during 0<t<T and a device that can measure the probability distribution of the particle's position at t=T. When a harmonic oscillator potential is applied at the instant t=0, the particle's wave function does not change at all during the rapid application process and remains as it is. And from then on, the particle undergoes harmonic oscillator motion with angular frequency ω. Its Hamiltonian is

Let us consider a physical quantity A as a linear combination of position and momentum.

The value of this A at t=0 can be determined from the value of x at t=T, x(T). x(T) is

given by, and using this, the value of A at t=0 is calculated as

This is the same not only in classical mechanics but also in quantum mechanics. At time t=T, let us consider the probability density distribution where the position x takes the value x=X.

Using this, the probability distribution of A at t=T is given by

In other words, the distribution at t=0 of the physical quantity A as an arbitrary linear combination of position x and momentum p can be measured.

Note that although Wigner made the incorrect argument in the last century that this A cannot be physically measured, in modern quantum measurement theory, A is understood as a physical quantity that can be measured without any problems.

Now, as a special case of A, using the angle variable θ

We will measure the probability distribution of the physical quantity at t=0 in this experiment. Using creation and annihilation operators, this can be written as

This is a physical quantity that often appears in fields such as quantum optics. Using this, let us construct a formula to determine the matrix elements of the density operator ρ at t=0 in the position representation from the experimental data of this physical quantity.

First, we define the Wigner function for ρ.

This is a real function that satisfies the normalization condition

Here, we consider the eigenvector where the physical quantity of interest takes the eigenvalue X.

Then, the probability density distribution obtained from ρ becomes

Through the method above, this is given as experimental data for each value of θ.

Now, using the delta function,

Since this relational expression holds, the probability density distribution in Equation (11) can also be written as

Also, the relational expression for the Fourier transform of the delta function is

Using this, equation (13) becomes

is calculated as follows. Here,

the definition of the trace, and the following Hausdorff formula

are used. At the same time,

recalling this, the following formula is obtained.

Therefore, using the Wigner function, the probability density of the measured physical quantity is

where,

if we set this as

it becomes,

leading to this formula. By performing a Fourier transform on both sides and then using an inverse Radon transform,

the Wigner function is uniquely determined from the experimental data. The formula for determining the density operator ρ from experimental data can also be obtained from this. First,

the relation

is obtained. Substituting Equation (24) yields

By substituting Equation (23) into this, it has been shown that the density operator ρ at time t=0 can be uniquely defined from the experimental data of the density distribution for each θ obtained by position measurement at time t=T. This serves as one method for quantum state tomography of a particle.

Furthermore, if the particle is a charged particle in two-dimensional or three-dimensional space, it is also possible to implement a similar quantum state tomography method experimentally by applying a magnetic field.