Superposition of two wavefunctions

[Leb]

[SOLVED] Superposition of two wavefunctions

Homework Statement

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

Homework Equations

[itex]|\psi|^{2}=\psi\psi^{*}[/itex]

The Attempt at a Solution

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

[itex]|\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)[/itex]

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

 

[spf]

Homework Statement

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

Homework Equations

[itex]|\psi|^{2}=\psi\psi^{*}[/itex]

The Attempt at a Solution

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

[itex]|\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)[/itex]

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

The following link should be helpful:

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

 

[Leb]

Great, thanks !

 

[spf]

Great, thanks !

Glad to help.

 

[asteriatic]

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.