Why is the derivative of a complex conjugate zero in quantum mechanics?

In summary: Partial derivatives with respect to z would be different from partial derivatives with respect to z^* because the former would involve derivatives with respect to z while the latter would involve derivatives with respect to the complex conjugate.
  • #1
daudaudaudau
302
0
Hi.

Sometimes in my quantum mechanics course we encounter derivatives such as [itex]\frac{d}{dz}z^*[/itex], i.e. the derivative of the complex conjugate of the complex variable z wrt z. We are told that this is just zero, even though I know that the complex conjugate is not an analytic function ... Has anyone else encountered this? I asked my teachers but they didn't know what was going on. Does this have anything to do with the fact that for an analytic function [itex]f(z)[/itex], the derivative wrt [itex]z^*[/itex] is zero, i.e. [itex]\frac{d}{dz^*}f(z)=0[/itex]?
 
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  • #2
daudaudaudau said:
Hi.

Sometimes in my quantum mechanics course we encounter derivatives such as [itex]\frac{d}{dz}z^*[/itex], i.e. the derivative of the complex conjugate of the complex variable z wrt z. We are told that this is just zero, even though I know that the complex conjugate is not an analytic function ... Has anyone else encountered this? I asked my teachers but they didn't know what was going on. Does this have anything to do with the fact that for an analytic function [itex]f(z)[/itex], the derivative wrt [itex]z^*[/itex] is zero, i.e. [itex]\frac{d}{dz^*}f(z)=0[/itex]?

The derivative of the complex conjugate isn't zero. It doesn't exist. Consider

[tex]\frac{f(z)-f(z_0)}{z-z_0}[/tex]

for [itex]f(z) = \overline z[/itex] and [itex]z_0 = 0[/itex]:

[tex]\frac{\overline z - \overline 0}{z - 0} = \frac{x-iy}{x+iy}[/tex]

Now [itex]z\rightarrow 0[/itex] won't give a limit at all as you can see by letting either x or y approach 0 first.
 
  • #3
Yeah, that's what I'm saying. The complex conjugate is not analytic (holomorphic). I'm just asking if anyone else have encountered the "trick" where you say that the derivative is just zero.

I've seen this used in quantum mechanics in connection with the variational principle. You expand the state vector on a basis and then you take the expectation value of the Hamiltonian in this state. Now you want the combination of coefficients that will give the lowest energy. To do this you differentiate the expectation value wrt the expansion coefficients, and when you encounter a conjugated coefficient, you just say that the derivative is zero. This will turn into a matrix equation in the expansion coefficients.

So the method works, I'm just asking why you can put the derivative of the conjugated expansion coefficients to zero.
 
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  • #4
I think you're dealing with something similar to this: [tex]\frac{\partial f}{\partial \overline{z} } = \frac{1}{2} \left( \frac{\partial f}{\partial x } +i \frac{\partial f}{\partial y } \right)[/tex] and so a function [tex]f[/tex] is analytic iff [tex]\frac{\partial f}{\partial \overline{z} }=0[/tex] . This is sometimes taken to mean that [tex]f[/tex] does not depend on [tex]\overline{z}[/tex] which would give a certain meaning to your equation (assuming of course that there is some other function [tex]f[/tex] involved).
 
  • #5
This is used in mathematics too. But only advanced mathematics. Instead of considering f a function of two variables x and y (where z = x+iy) we can consider f a function of two variables z and [itex]z^*[/itex] (or, as mathematicians say, z and [itex]\overline{z}[/itex] ). So now we do partial derivatives with respect to these two variables. Please note that partial derivative [itex]\partial/\partial z[/itex] is NOT THE SAME as ordinary derivative [itex]d/dz[/itex]. The Cauchy-Riemann equations become [itex]\partial f/\partial \overline{z} = 0[/itex], very nice.

Should physicists who don't know the mathematics avoid using it? Try telling that to physicists! It has no effect!
 
  • #6
g_edgar said:
Please note that partial derivative [itex]\partial/\partial z[/itex] is NOT THE SAME as ordinary derivative [itex]d/dz[/itex].

That's a good point. I was thinking about analyticity in terms of the total derivative.
 
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Related to Why is the derivative of a complex conjugate zero in quantum mechanics?

What is a complex conjugate derivative?

A complex conjugate derivative is a type of derivative that is calculated using the complex conjugate of a function. It is used in complex analysis to find the rate of change of a function with respect to its complex variable.

Why is the complex conjugate used in the derivative?

The complex conjugate is used in the derivative because it allows for the differentiation of functions with complex variables while still following the rules of traditional calculus. It also helps to simplify the calculation of the derivative in certain cases.

How is the complex conjugate derivative calculated?

The complex conjugate derivative is calculated using the standard rules of differentiation, but with the added step of taking the complex conjugate of the function. This means that if the function is written as f(z), the complex conjugate derivative would be written as f*(z).

What are some applications of the complex conjugate derivative?

The complex conjugate derivative has various applications in physics, engineering, and mathematics. It is commonly used in the study of electromagnetic fields, quantum mechanics, and signal processing. It is also used in the calculation of integrals and finding the solutions to differential equations.

Are there any limitations to using the complex conjugate derivative?

One limitation of using the complex conjugate derivative is that it only applies to functions with complex variables. It cannot be used for functions with only real variables. Additionally, the use of complex conjugate can sometimes make the calculations more complicated and difficult to understand.

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