# Mathematical conundrum when adding complex exponentials

• Runei
In summary, the solution to the infinite square well can be written as either a complex exponential or a sine and cosine, but the solution with complex exponentials does not satisfy the constraints of the well.
Runei
Hi there,

Once again I find myself twiddling around with some quantum mechanics, and I bumped into something I find strange. I can't see what the error of my thinking is, so I hope someone could be able to point it out.

I'm looking at solutions to the infinite square well, and arrive at the simple differential equation

$$\frac{d^2\Psi}{dx^2} = -k^2 \Psi$$

The solution to this can be written in terms of complex exponentials or sines and cosines. I bumped into the weird stuff when I use complex exponentials.

So the general solution in that case would be

##\Psi(x) = Ae^{ikx}+Be^{-ikx}##

Now, what I then started thinking was: "Hmmm... This could be viewed mathematically as a sum of two vectors, and solution is simply another vector."

So I drew this picture to illustrate the idea:

So from that perspective it seems that the solution could also be written as

##\Psi(x) = Ce^{ik'x}##

However, using the simple constraints of the infinite square well quickly leads to problems - namely:

##|\Psi(0)|^2 = 0##

##C = 0##

So... Now really what I had hoped for. Where am I going taking a wrong turn? Has it to do with the x? That x should be x' also?

#### Attachments

• drawing.png
6.7 KB · Views: 647
The problem is that your ##k'## is going to be a function of ##x##. As such, you can not assume it to be constant and apply derivatives to the wave function with that assumption.

bhobba and Runei
Consider that ##e^{ix} + e^{-ix}## is equal to ##2 cos(x)## and so has no net imaginary component for all x. But a single ##e^{kix}## always has some net imaginary component (for finite ##k## and ##x##).

Thanks guys!

I gave it some more thought and think I nailed it down now.

I have another question now thought, but I'm going to make another thread for it.

## 1. What is a complex exponential?

A complex exponential is a mathematical expression that involves a complex number raised to a power. It is written in the form z = a + bi, where a and b are real numbers and i is the imaginary unit √(-1).

## 2. What is a mathematical conundrum?

A mathematical conundrum is a problem or puzzle that is difficult to solve or understand. In the case of adding complex exponentials, it refers to the challenge of simplifying the expression and finding the correct solution.

## 3. How do you add complex exponentials?

To add complex exponentials, you must first convert them to polar form. Then, you can add the magnitudes and add the angles to find the final solution. Finally, convert the result back to rectangular form if necessary.

## 4. What is the Euler's formula and how is it related to complex exponentials?

Euler's formula is e^(ix) = cos(x) + i*sin(x), where e is the base of the natural logarithm and i is the imaginary unit. This formula is related to complex exponentials because it shows the relationship between the trigonometric functions and the exponential function.

## 5. What are some real-world applications of complex exponentials?

Complex exponentials have many applications in fields such as engineering, physics, and signal processing. They are used to describe phenomena with both real and imaginary components, such as alternating current in electrical circuits or the oscillatory behavior of waves. They are also used in the solution of differential equations and in the study of quantum mechanics.

• Quantum Physics
Replies
21
Views
2K
• Quantum Physics
Replies
1
Views
876
• Quantum Physics
Replies
0
Views
425
Replies
10
Views
787
• Quantum Physics
Replies
8
Views
2K
• Quantum Physics
Replies
14
Views
1K
• Quantum Physics
Replies
1
Views
911
• Quantum Physics
Replies
17
Views
2K
• Quantum Physics
Replies
12
Views
2K
• Quantum Physics
Replies
2
Views
722