Derivative of an imaginary exponential

In summary, the conversation discusses how to take the derivative of e^((x^2)/i) and the use of substitution to simplify the problem. The participants also mention the difficulty of applying the substitution to a more complex problem involving an expectation value for momentum. They also mention seeking help from a TA to solve the issue.
  • #1
Shock
14
0
Does anyone know how to take the derivative of e^((x^2)/i)?

Thanks in advance!
 
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  • #2
The derivative of eu is eudu

If u = x^2/i, find du. Substitute u and du, and you're done.

- Warren
 
  • #3
Bleh ya, alright, i was hoping there was a shortcut. The actual problem is much nastier, and the u substitution is giong to be hard to apply throughout.

Thanks man!
 
  • #4
It isn't a very lengthy procedure.. you just have to find the derivative of u=x^2/i, which should take you only a minute, then plug it into eudu.

Do you need help with the actual problem?

- Warren
 
  • #5
ya actually that would be good, I am still getting an imaginary left over in my <p>.

I'm trying to take a expectation value for momentum, the operator is h/i(d/dx) (hbar that is) and the actual thing I am taking the derivative of is this

e^(-(ax^2)/(1+2ihat/m)) (all constants except for x) and doing what you just told me i get:

du=(-2am^2x)/(4a^2h^2t^2+m^2)+(4ia^2hmxt)/(4a^2h^2t^2+m^2)

there is also a sqrt(1+2hiat/m) under the exponential, but i want to leave that alone so i can take the intergral of x*(wavefn)^2, (the wavefn^2 i calculated in a different part of the problem, and a constant w came out of it that i need to keep everythign in terms of for this part)

So I am left with an i in my expectation value for my momentum, which is wrong :(
 
Last edited:
  • #6
Are you taking the derivative of u with respect to x or t or m? I can't see how you got your du.

- Warren
 
  • #7
x, and my calculator did it :D
 
  • #8
it hasent failed me in the past, i can try doing it out by hand though to be sure
 
  • #9
actually, i have no clue why it got all that
 
  • #10
well in any case I am still stuck with an imaginary momentum
 
  • #11
apparently <p^2>=a*(hbar)^2 LOL maybe ill try and work towards that, and see what to do, it says itll take a lot of alegbra to get it to that form though!
 
  • #12
still looking like an i is going to follow through, ill just go annoy the gsi (TA) about it, thanks for your help tho
 

FAQ: Derivative of an imaginary exponential

What is the derivative of an imaginary exponential?

The derivative of an imaginary exponential is the product of the imaginary unit (i) and the original function. For example, if the function is f(x) = e^(ix), the derivative would be if(x) = i(e^(ix)). This is because the derivative of e^(ix) is i(e^(ix)) and the derivative of i is -1.

How is the derivative of an imaginary exponential calculated?

The derivative of an imaginary exponential can be calculated using the chain rule. This means that the derivative of e^(ix) is calculated by multiplying the derivative of the function inside the parentheses (i.e. ix) by the derivative of the function outside the parentheses (i.e. e^(x)).

What is the general form of the derivative of an imaginary exponential?

The general form of the derivative of an imaginary exponential is if(x) = i(e^(ix)) where i is the imaginary unit and e^(ix) is the original function.

Why is the derivative of an imaginary exponential important in mathematics?

The derivative of an imaginary exponential is important in mathematics because it helps us calculate the rate of change of these types of functions. It is also used in various mathematical applications, such as differential equations and Fourier analysis.

Can the derivative of an imaginary exponential be simplified?

Yes, the derivative of an imaginary exponential can be simplified by using the properties of the imaginary unit (i). For example, if the function is f(x) = e^(ix), the derivative would be if(x) = i(e^(ix)). However, this can be simplified to f(x) = -sin(x) + icos(x) by using the trigonometric identities e^(ix) = cos(x) + isin(x) and i^2 = -1.

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