Wave Function Doubt and Derivation

[sarvesh0303]

Homework Statement

I was reading up on the Wave Function used in the Schrodinger Wave Equation. However one source said that
ψ(x,t)=e^(-i/hbar*(px-Et))
Another source had this
ψ(x,t)=e^(i/hbar*(px-Et))

Which one of these is true and could someone give a derivation for the correct wavefunction?

Homework Equations

The Attempt at a Solution

 

[Simon Bridge]

You can see for yourself by substituting into the 1DSE.
In fact - that is an important exercise: do that and get back to us.

 

[sarvesh0303]

I know I can! But I want to try deriving the 1DSE by using this value. So I want to know another way of deriving it. I read that it can be derived by taking
ψ(x,t)=A(cos(2∏x/λ-2∏ηt)+isin(2∏x/λ-2∏ηt))

where η=frequency
and then the euler formula was used.

My doubt with this derivation is that would every wave (its wavefunction) always be of the form mentioned above.
If so then since isin(2∏x/λ-2∏ηt) is a complex term, then wouldn't it imply that every wave must have a complex component?
This part really confuses me!

 

[Simon Bridge]

I know I can! But I want to try deriving the 1DSE by using this value.

The 1DSE is not really something you derive is it? You will benefit by substituting both forms of the wavefunction you were asking about into the Schrodinger equation to see which one is the "real" solution. Have you tried this yet? If not - do it. If you have, please report what you found.

So I want to know another way of deriving it.

"it" what? The wavefunction for a free particle or the 1DSE? You have mentioned trying to derive both now.

My doubt with this derivation is that would every wave (its wavefunction) always be of the form mentioned above.
If so then since isin(2∏x/λ-2∏ηt) is a complex term, then wouldn't it imply that every wave must have a complex component?
This part really confuses me!

In general, wave-functions are complex - this is correct. Why would this confuse you? It is why you have to premultiply by the complex conjugate before integrating.

 

[Nickie]

I understand your concern regarding the discrepancy between the two sources. However, both of these wave functions are valid and can be derived from the Schrodinger Wave Equation.

The difference between the two sources is due to the choice of the sign convention for the imaginary unit (i). Some sources use the convention where i is defined as e^(-iπ/2), while others use the convention where i is defined as e^(iπ/2). This results in the difference in the overall sign of the wave function.

To derive the correct wave function, we can start with the Schrodinger Wave Equation:

iħ∂ψ/∂t = -ħ^2/2m ∂^2ψ/∂x^2 + V(x)ψ

Where ħ is the reduced Planck's constant, t is time, m is the mass of the particle, x is position, V(x) is the potential energy, and ψ is the wave function.

To solve this equation, we can use the method of separation of variables, where we assume that ψ(x,t) can be written as a product of two functions, one dependent on x and the other dependent on t:

ψ(x,t) = X(x)T(t)

Substituting this into the Schrodinger Wave Equation, we get:

iħT(t)dX(x)/dx = -ħ^2/2m X(x)d^2T(t)/dt^2 + V(x)XT(t)

Dividing both sides by ψ(x,t) = X(x)T(t), we get:

iħ/T(t) dT(t)/dt = -ħ^2/2m X(x)/T(t) d^2X(x)/dx^2 + V(x)

Since the left side of the equation only depends on t and the right side only depends on x, both sides must be equal to a constant, which we can denote as E. This constant represents the total energy of the particle.

iħdT(t)/dt = ET(t)

And

-ħ^2/2m d^2X(x)/dx^2 + V(x)X(x) = EX(x)

We can solve the first equation for T(t) to get:

T(t) = e^(-iEt/ħ)

Substituting this into the second equation, we