# Many particle wavefunction

• Brewer
In summary, the conversation discusses the normalization of a wavefunction for two non-interacting particles, specifically for bosons. The formula for the wavefunction is given, and it is mentioned that in order to normalize it, the modulus squared must be found. However, the issue arises when no wavefunctions are given in the exam questions and the answer is still given in terms of phi 1 and phi 2. The speaker is unsure of how to find the modulus squared without knowing if the answer is complex, and how to integrate without substituting values for the wavefunctions. It is suggested that the single-particle wavefunctions may already be normalized, making the explicit integration unnecessary.

#### Brewer

If in the case of two non-interacting particles, the wavefunction looks like (for bosons):

$$\psi(x_1, x_2) = \frac{1}{\sqrt{2}}[\phi_a(x_1)\phi_b(x_2) + \phi_a(x_2)\phi_b(x_1)$$

And to normalise the wavefunction, the modulus squared has to be found. I can do this when I can substitute standard wavefunctions into the equations (either harmonic oscillator, or a square well for example), but I've been looking at exam papers and in questions with this kind of question, no wavefunctions are given and the answer (its normally a "show that..." question) is still given in terms of phi 1 and phi 2.

I'm unsure how the modulus squared bit works when I don't know for sure that the answer is complex. I also don't know how to integrate these without substituting values in for the wavefunctions. Any help going through this would be appreciated.

It's been a long time since I've done any many-particle qm. But I can tell you that you definitely cannot integrate without knowing the wavefunctions. However, there may be some mention that the single-particle wavefunctions (i.e. the phi's) are themselves normalized. Then you know that e.g.
$$\int |\phi_a|^2 dx = 1$$, so you likely don't have to do any integrals explicitly.

## What is a many particle wavefunction?

A many particle wavefunction is a mathematical description of the quantum state of a system composed of multiple interacting particles. It is a complex-valued function that describes the probability of finding the particles in a specific configuration in space and time.

## Why is the many particle wavefunction important?

The many particle wavefunction is important because it allows us to understand and predict the behavior of quantum systems, which is crucial in fields such as quantum chemistry, solid state physics, and nuclear physics. It also plays a central role in the development of quantum computing and communication technologies.

## How is the many particle wavefunction different from the single particle wavefunction?

The many particle wavefunction is a function of all the coordinates and momenta of all the particles in the system, while the single particle wavefunction only depends on the coordinates and momenta of a single particle. In addition, the many particle wavefunction is a multi-dimensional function, whereas the single particle wavefunction is a one-dimensional function.

## What is the role of symmetry in the many particle wavefunction?

Symmetry plays an important role in the many particle wavefunction as it determines the behavior of the particles and their interactions. The wavefunction must be symmetric or anti-symmetric depending on the properties of the particles (e.g. bosons or fermions), in order to satisfy the Pauli exclusion principle and other fundamental principles of quantum mechanics.

## How is the many particle wavefunction used in experiments?

The many particle wavefunction is used to calculate the probabilities of different outcomes in physical experiments involving quantum systems. It can also be used to simulate and analyze the behavior of complex systems, providing insights into their properties and behavior. In addition, it is used to design and optimize experiments in fields such as quantum information and quantum technology.