Quantum Mechanics (Wien’s displacement law)

[says]

Homework Statement

Show that the maximum of the Planck energy density occurs for a wavelength of the form λ_max = b/T, where T is the temperature and b is a constant that needs to be estimated.

Homework Equations

Planck energy density
u (v,T) = 8πv² / c³ * hv / e^hv/kT-1

The Attempt at a Solution

v= c / λ
dv = |dv / (dλ)| dλ = (c/λ²) dλ

I get up to this bit and I'm stuck...

v= c / λ
c = v*λ
∴ v = v*λ / λ
dv = dv*dλ / dλ
dv = (dv / dλ) * dλ

I'm confused how dv = |dv / (dλ)| dλ turns into (c/λ²) dλ. Sorry if my question is a bit vague -- haven't posted on here in a while and I just started taking QM.

 

[ehild]

Homework Statement

Show that the maximum of the Planck energy density occurs for a wavelength of the form λ_max = b/T, where T is the temperature and b is a constant that needs to be estimated.

Homework Equations

Planck energy density
u (v,T) = 8πv² / c³ * hv / e^hv/kT-1

The Attempt at a Solution

v= c / λ
dv = |dv / (dλ)| dλ = (c/λ²) dλ

I get up to this bit and I'm stuck...

v= c / λ
c = v*λ
∴ v = v*λ / λ
dv = dv*dλ / dλ
dv = (dv / dλ) * dλ

I'm confused how dv = |dv / (dλ)| dλ turns into (c/λ²) dλ. Sorry if my question is a bit vague -- haven't posted on here in a while and I just started taking QM.

The absolute value marks are wrong in your equation.
d means differential, and dv/dλ is the derivative of v with respect to λ. d is not a multiplicative factor!
v is function of λ, and u(v,T)=F(v(λ),T). v=c/λ(v). You have to apply the chain rule to find the λ, where the u( λ) plot has its maximum:
dF/dλ = df/dv dv/dλ.
v=c/ λ, what is its derivative with respect to λ?

 

[says]

v=c/ λ, what is its derivative with respect to λ?

dv = ( c / λ² )

 

[says]

v=c/ λ, what is its derivative with respect to λ?
dv = ( c / λ² ) dλ

 

[ehild]

v=c/ λ, what is its derivative with respect to λ?

dv = ( c / λ² )

No, the derivative is written as $\frac{dv}{dλ}$
In the derivative, you miss a minus sign.