Replacing total derivative with partial derivative in Griffiths' book

In summary, Griffiths is saying that when integrating a function between limits, you should use the partial derivative since the resulting function doesn't depend on x as well as t. When taking the time derivative of this function, you should use the total derivative since the function depends on x and t.
  • #1
I'm using Griffiths' book to self-study QM and I'm having a slight problem following one of his equations. In page 11 of his "Intro to Quantum Mechanics (2nd ed.)", he gives the reader the following 2 equations:

[tex] \frac {d} {dt} \int_{-\infty}^{\infty}|\Psi(x,t)|^2 dx = \int_{-\infty}^{\infty} \frac{\partial}{\partial t} |\Psi(x,t)|^2 dx [/tex]

In the next line, he gives the following explanation:
"Note that the integral is a function of only t, so I use the total derivative (d/dt) in the first term, but the integrand is a function of x as well as t, so it's a partial derivative ([tex]\frac{\partial}{\partial t}[/tex]) in the second one."

It is this explanation that I am having difficulty understanding. I just don't understand why he replaced the total derivative with a partial derivative, despite his explanation. Could someone please explain what he is trying to explain in greater detail? Thanks.
 
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  • #2
Suppose we integrate some function of x between limits, then after performing the integration, we are left with a number (since we have evaluated x in the resulting expression at the limits). Similarly, if we have a function of x and t, like above, then when integrating it with respect to x, we will be left with a function of only t (since, again, x has been evaluated at the limits). So, the derivative of this wrt t will be a total derivative.

Now consider the second case above, where we are taking the time derivative into the integral. Now, the function inside the integral is a function of x and t (since it has not been integrated yet) and so the derivative wrt t will be partial.
 
  • #3
In the LHS a function of both "x" and "t" is integrated DEFINTELY wrt "x" resulting in a function only of "t". The "x" dependent part has been replaced with the limits of the antiderivative at the points + and - infinity. Therefore, the resulting function doesn't depend on 2 variables anymore and depends on "t" wrt which is differentiated.

Since "x" and "t" are independent variables, it doesn't matter whether you integrate wrt "x" first and differentiate wrt "t" afterwards (LHS) or viceversa (RHS). The reason we use the partial derivative in the RHS is because the probability density is a function of 2 independent variables and differentiation wrt one is denoted by the Jacobi symbol [itex] \partial [/itex].
 
  • #4
*thumbs up*

Thanks guys, it's much clearer to me now :biggrin:
 

1. What is the difference between total derivative and partial derivative?

The total derivative is used to calculate the change in a function with respect to all of its independent variables simultaneously, while the partial derivative is used to calculate the change in a function with respect to only one of its independent variables, holding the others constant.

2. Why does Griffiths use partial derivatives instead of total derivatives in his book?

Griffiths uses partial derivatives because most of the physical systems he discusses in his book involve multiple independent variables, and it is more convenient to use partial derivatives to represent the relationship between these variables.

3. Are there any cases where total derivatives are more appropriate to use than partial derivatives?

Yes, there are some cases where using total derivatives is more appropriate. For example, if the independent variables are not related to each other, or if the function being studied is only dependent on one variable, then total derivatives would be more suitable.

4. Can total derivatives and partial derivatives be used interchangeably?

No, total derivatives and partial derivatives are two different concepts and cannot be used interchangeably. It is important to use the correct type of derivative depending on the specific situation and variables involved.

5. How does replacing total derivatives with partial derivatives affect the accuracy of calculations?

Replacing total derivatives with partial derivatives does not significantly affect the accuracy of calculations, as long as the variables involved are properly accounted for and the correct type of derivative is used for each situation. However, in some cases, using total derivatives may provide a more accurate representation of the system being studied.

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