# Superposition of two wavefunctions

• Leb
In summary, the conversation involves the solution to a problem with complex number arithmetic and the application of a given equation. The final result is compared to notes and a helpful link is provided.
Leb
[SOLVED] Superposition of two wavefunctions

## Homework Statement

The problem is more of complex number arithmetic more then conceptual :

## Homework Equations

$|\psi|^{2}=\psi\psi^{*}$

## The Attempt at a Solution

I simply used the equation given above, but instead of getting 2Re{...} I get :

$|\psi_{1}||\psi_{2}| \left( c_{1}c_{2}^{*}\exp({i(\alpha_{1}-\alpha_{2})})+c_{1}^{*}c_{2}\exp({-i(\alpha_{1}-\alpha_{2})})\right)$

Could someone explain how is this equal to that given in the notes I attached ?

Last edited:
Leb said:

## Homework Statement

The problem is more of complex number arithmetic more then conceptual : View attachment 44717

## Homework Equations

$|\psi|^{2}=\psi\psi^{*}$

## The Attempt at a Solution

I simply used the equation given above, but instead of getting 2Re{...} I get :

$|\psi_{1}||\psi_{2}| \left( c_{1}c_{2}^{*}\exp({i(\alpha_{1}-\alpha_{2})})+c_{1}^{*}c_{2}\exp({-i(\alpha_{1}-\alpha_{2})})\right)$

Could someone explaind how is this equal to that given in the notes I attached ?

http://en.wikipedia.org/wiki/Complex_number#Conjugation

Great, thanks !

Leb said:
Great, thanks !

The superposition of two wavefunctions is a fundamental concept in quantum mechanics. It refers to the combination of two or more wavefunctions to describe a quantum system. The resulting wavefunction is a linear combination of the individual wavefunctions, with coefficients representing the probability amplitudes for each state.

In mathematical terms, the superposition of two wavefunctions, \psi_1 and \psi_2, can be represented as:

\psi = c_1\psi_1 + c_2\psi_2

where c_1 and c_2 are complex coefficients. The probability density of this combined wavefunction is given by:

|\psi|^2 = |c_1\psi_1 + c_2\psi_2|^2

Using the equation |\psi|^{2}=\psi\psi^{*}, we can expand this to:

|\psi|^2 = (c_1\psi_1 + c_2\psi_2)(c_1^*\psi_1^* + c_2^*\psi_2^*)

= c_1c_1^*\psi_1\psi_1^* + c_1c_2^*\psi_1\psi_2^* + c_2c_1^*\psi_2\psi_1^* + c_2c_2^*\psi_2\psi_2^*

= |c_1|^2|\psi_1|^2 + |c_2|^2|\psi_2|^2 + c_1c_2^*\psi_1\psi_2^* + c_1^*c_2\psi_2\psi_1^*

Using the properties of complex numbers, we can simplify this to:

|\psi|^2 = |c_1|^2|\psi_1|^2 + |c_2|^2|\psi_2|^2 + 2Re(c_1c_2^*\psi_1\psi_2^*)

This is the same result as the one given in the notes. Therefore, the equation you used is correct and equivalent to the one in the notes.

## 1. What is the definition of superposition of two wavefunctions?

The superposition of two wavefunctions is a fundamental concept in quantum mechanics where two or more wavefunctions are combined to produce a new wavefunction. This new wavefunction describes the probability of finding a particle in a particular state or location.

## 2. How is the superposition of two wavefunctions calculated?

The superposition of two wavefunctions is calculated by adding or subtracting the individual wavefunctions. The resulting wavefunction is a combination of the amplitudes and phases of the original wavefunctions.

## 3. What is the significance of superposition in quantum mechanics?

The concept of superposition is crucial in understanding the behavior of particles at the quantum level. It explains how particles can exist in multiple states simultaneously and how their probabilities can be calculated.

## 4. Can superposition of two wavefunctions occur in classical physics?

No, the superposition of two wavefunctions is a concept unique to quantum mechanics. In classical physics, particles are described by definite states and cannot exist in multiple states at the same time.

## 5. What are some real-life applications of superposition of two wavefunctions?

The superposition of two wavefunctions has many practical applications, such as in quantum computing, where qubits (quantum bits) can exist in multiple states simultaneously. It is also used in technologies like MRI machines and atomic clocks. Additionally, the principle of superposition is being explored for potential applications in communication and encryption.

• Introductory Physics Homework Help
Replies
4
Views
365
• Introductory Physics Homework Help
Replies
12
Views
421
• Quantum Physics
Replies
5
Views
682
• Quantum Physics
Replies
2
Views
752
• Introductory Physics Homework Help
Replies
7
Views
252
• Introductory Physics Homework Help
Replies
14
Views
2K
• Introductory Physics Homework Help
Replies
9
Views
1K
Replies
1
Views
1K
• Introductory Physics Homework Help
Replies
29
Views
2K
• Quantum Physics
Replies
8
Views
2K