Question from Griffiths book?

In summary, the equation (1.21) in the normalization chapter involves a total derivative in the first expression, but since the integrand is a function of both x and t, a partial derivative is used in the second expression. This is because the integral is a function of t only and the derivative can be taken with respect to t. However, to account for the fact that the integration range does not change with time, the time derivative is taken at a fixed x value, requiring the use of a partial derivative.
  • #1
In a normalization chapter there's an equation(1.21) which says: d/dt ∫|ψ(x,t)|[itex]^{2}[/itex]dx=∫∂/∂t |ψ(x,t)|[itex]^{2}[/itex]dx
there was a description:(Note that integral is a function only of t,so I use a total derivative (d/dt) in the first expression,but the integrand is a function of x as well as t , so it's a partial derivative in the second one (∂/∂t) )
so this textbook started very simple and intuitive, but now I'm really confused.First of all why did d/dt appear and why did it "transform" to a partial derivative as it "entered" the integral?
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  • #2
The integral
[tex]N(t)=\int_{\mathbb{R}} \mathrm{d} x \; |\psi(x,t)|^2[/tex]
is a function of [itex]t[/itex] only, because you integrate over [itex]x[/itex]. Thus you can take only the derivative wrt. [itex]t[/itex].

Now, because the integration range [itex]\mathbb{R}[/itex] doesn't change with time, you can as well take first the time derivative of the integrand and then integrate wrt. [itex]x[/itex]. Now, since you have a function of [itex]x[/itex] and [itex]t[/itex] you must indicate that you take the time derivative at fixed [itex]x[/itex]. That's why you have to use a partial derivative:
[tex]\frac{\mathrm{d}N(t)}{\mathrm{d} t} = \int_{\mathbb{R}} \mathrm{d} x \frac{\partial}{\partial t} |\psi(x,t)|^2.[/tex]
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1. What is the main topic of Griffiths book?

The main topic of Griffiths book is introductory quantum mechanics.

2. Who is the target audience for Griffiths book?

The target audience for Griffiths book is undergraduate students in physics or related fields.

3. What sets Griffiths book apart from other introductory quantum mechanics textbooks?

Griffiths book is known for its clear and concise explanations, use of real-world examples, and emphasis on conceptual understanding rather than mathematical formalism.

4. Is Griffiths book suitable for self-study?

Yes, Griffiths book is often used for self-study as it is well-organized and includes numerous practice problems and solutions.

5. Are there any prerequisites for reading Griffiths book?

Basic knowledge of calculus and linear algebra is recommended for understanding the concepts in Griffiths book.

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