# Can this function be a wavefunction of a physical system?

• td21
In summary, the function \psi(x)=\frac{A}{\sqrt{x^2+b^2}} can be a wavefunction for a physical system with finite potential energy. This is supported by the fact that the function is continuous and its derivative is also continuous. Additionally, the function can be normalized as it approaches zero as x approaches infinity and is finite when x=0. To prove this, integration is necessary to show that A is finite. However, it is important to consider the potential being time independent and whether it can be a solution to the spectral equation of the Hamiltonian.
td21
Gold Member

## Homework Statement

Can this function be a wavefunction of a physical system with finite potention energy:
$$\psi(x)=\frac{A}{\sqrt{x^2+b^2}}$$

no

## The Attempt at a Solution

The ans is YES.
1)it is continuous.
2)its derivative also continous.
3)It can be normalized, as it tends to zero as x tends to infinity. Also when x=0, this function is finite.Is the above argument enough? Also for 3), do i need to do an integration to prove it can be normalized? Thanks!

Assuming the potential to be time independent, can it be a solution to the spectral equation of the Hamiltonian ?

for 3), you need to do the integration and show that A is finite.

## 1. Can any mathematical function be a wavefunction of a physical system?

No, not all mathematical functions can serve as wavefunctions for a physical system. The wavefunction must satisfy certain mathematical criteria, such as being continuous and square-integrable, in order to accurately describe the behavior of a physical system.

## 2. How do I know if a given function is a valid wavefunction for a physical system?

To determine if a function is a valid wavefunction for a physical system, it must satisfy the Schrödinger equation, which is a differential equation that describes the behavior of quantum systems. Additionally, the wavefunction must be normalized and have a finite value everywhere in space.

## 3. Can a wavefunction change over time?

Yes, the wavefunction of a physical system can change over time according to the Schrödinger equation. This change is known as wavefunction evolution and describes how the probability distribution of a quantum system changes over time.

## 4. What is the relationship between the wavefunction and the physical properties of a system?

The wavefunction is a mathematical representation of the physical state of a quantum system. It contains information about the system's position, momentum, and other physical properties. The square of the wavefunction, known as the probability density, provides information about the likelihood of finding the system in a particular state.

## 5. Is the wavefunction a real physical entity?

No, the wavefunction is a mathematical construct used to describe the behavior of quantum systems. It does not have a physical reality in the traditional sense, but it represents the probability of a system's physical properties and is an essential tool for understanding and predicting the behavior of quantum systems.

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