# Creating Wave Functions With One Variable: Can It Be Done?

• johann1301
In summary, there is a general equation for waves that is a second order differential equation. It is possible to create a wave function with just one variable using the number i, such as f(x)=i^x or f(x)=2^ix. However, it is also possible to construct a one-dimensional wave of any shape without using the number i, using the expression r(x)e^(ig(x)). If you want a 'real' waveform with the same peak and trough for every wavefront, you can use r(x) = c. However, this would be considered "cheating" as it still involves the use of the imaginary unit. So, it is not possible to create a wave function without using the number i AND with
johann1301
With the number i, its possible to create a wave function with just one variable.

example:

f(x)=i^x

or..

f(x)=2^ix

But is there any wave function without the number i AND with only one(x and y) variable? can this be created?

Last edited:
y = sin(x)

Hello,

Are you aware of the general equation of a wave?

Hint: (It is a second order differential equation)

johann1301 said:
With the number i, its possible to create a wave function with just one variable.

example:

f(x)=i^x

or..

f(x)=2^ix

But is there any wave function without the number i AND with only one(x and y) variable? can this be created?

If you want something that oscillates, then something of the form r(x)e^(ig(x)) will give you that. A standard linear wave will be in the form g(x) = wx + c corresponding to a shift and frequency information and if you want a 'real' waveform with the same peak and trough for every wavefront then r(x) = c.

This expression will give you a complex wave-form with a real and imaginary component and if you want one or the other and not both you need to use the normal identities to get rid of the i term or the real term.

In fact, it is possible to construct a one-dimensional wave of any shape.

y = sin(x)

This is the same as

y=1/2 i e^(-i x)-1/2 i e^(i x)

so that would be "cheating"...

Last edited:
(ig(x))

does this mean i*g(x) (the imaginary unit times the function g of x)?

Hello,

Are you aware of the general equation of a wave?

Hint: (It is a second order differential equation)

I don't know anything about differential equations:(

## 1. Can we create wave functions with just one variable?

Yes, it is possible to create wave functions with just one variable. These are known as one-dimensional wave functions, and they are commonly used in quantum mechanics to describe the behavior of particles in one-dimensional systems.

## 2. What is the mathematical representation of a one-dimensional wave function?

A one-dimensional wave function is typically represented as a complex-valued function of one variable, usually denoted as Ψ(x). This function describes the probability amplitude of a particle being at a certain position x in the one-dimensional system.

## 3. Can one-dimensional wave functions accurately describe real-world systems?

Yes, one-dimensional wave functions can accurately describe certain real-world systems, such as particles moving along a straight line or a standing wave on a string. However, for more complex systems, multiple variables may be needed to fully describe the wave function.

## 4. How do we create a one-dimensional wave function?

One-dimensional wave functions can be created through mathematical methods, such as solving the Schrödinger equation for a one-dimensional system. They can also be constructed using known physical laws and experimental data.

## 5. What is the significance of one-dimensional wave functions in quantum mechanics?

One-dimensional wave functions play a crucial role in quantum mechanics as they provide a mathematical description of the behavior of particles in one-dimensional systems. They allow us to make predictions about the behavior of particles and understand the fundamental principles of quantum mechanics.

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