# Proving Complex Wave Function: |ψ1 + ψ2|^2 in [0, 4α]

• jc09
In summary, the conversation discusses the problem of showing that for two complex functions, ψ1 and ψ2, both belonging to the set of complex numbers, the absolute value squared of their sum can take any value within the range of 0 to 4 times the modulus of ψ1 squared, which is equal to the modulus of ψ2 squared. The solution proposed involves using the specific forms of the functions and their respective moduli to solve the problem, without the need for the triangle inequality.
jc09
Show that for ψ1 , ψ2 ∈ C,
|ψ1 + ψ2 |^2
can take any value in [0, 4α], where
α = |ψ1 |^2 = |ψ2 |^ 2

I think the solution has something to do with triangular identies but I am not sure how to start this problem at all.

I think you can get away without using the triangle inequality. If $\psi_1$ and $\psi_2$ both have constant modulus $\sqrt{a}$ then the functions must be of the following form (assuming 1 spatial dimension and time independence).

$$\psi_1(x)=\sqrt{a}\exp\left(ik_1x+i\phi_1\right)$$

$$\psi_1(x)=\sqrt{a}\exp\left(ik_2x+i\phi_2\right)$$

Here $k_1,k_2,\phi_1,\phi_2\in\mathbb{R}$.

Can you do something with that?

## 1. What is a complex wave function?

A complex wave function is a mathematical representation of a wave that includes both magnitude and phase information. It is commonly used in quantum mechanics to describe the behavior of particles.

## 2. What does |ψ1 + ψ2|^2 represent?

|ψ1 + ψ2|^2 is the probability amplitude, or the squared magnitude, of the sum of two complex wave functions. It represents the probability of finding a particle in a particular state when both wave functions are present.

## 3. What is the significance of [0, 4α] in the equation?

The [0, 4α] interval represents the spatial region in which the complex wave function is being measured. It can be thought of as the "space" in which the particle is moving and where its probability amplitude is being calculated.

## 4. How can we prove the equation |ψ1 + ψ2|^2 in [0, 4α]?

The equation can be proved through mathematical manipulation and the application of quantum mechanics principles. This may involve using the Schrödinger equation, boundary conditions, and other mathematical techniques to show that the equation holds true for a given range of values.

## 5. What is the physical interpretation of |ψ1 + ψ2|^2?

The physical interpretation of |ψ1 + ψ2|^2 is the probability of finding a particle in a particular state within the specified interval [0, 4α]. This interval can represent the position, momentum, or energy of the particle, depending on the type of complex wave function being used. The higher the value of |ψ1 + ψ2|^2, the greater the likelihood of finding the particle in that state.

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