Given this formula, is this formula true by symmetry?

• bjnartowt
In summary, the commutator [f(x),p] is equal to i*h*df/dx, while [f(p),x] is equal to i*h*df/dp. However, this is not simply due to symmetry as the symbols in quantum mechanics hold specific meanings and can change based on their associations with other operators. Operators that commute are highly important in quantum mechanics as they do not change the overall meaning of the operation.
bjnartowt

Homework Statement

I know that,
$$\left[ {f(x),p} \right] = {\bf{i}}\hbar \frac{{df}}{{dx}}$$

By symmetry, is it also true that,
$$\left[ {f(p),x} \right] = {\bf{i}}\hbar \frac{{df}}{{dp}}$$

...since x and p are just symbols?

No. These are commutators, and the symbols aren't just placeholders, they actually mean something. Almost everything in quantum mechanics has a meaning that can change based on how you associate that meaning to something else. That is why operators that commute have such a high importance, the overall meaning of what you are doing doesn't change between the two operators that commute.

$$[f(x),p]g=-i*h*\frac{d}{dx}(f*g)+ihf*dg/dx = -ih(gdf/dx+fdg/dx-fdg/dx)= ihgdf/dx$$

The g's go away and you get your answer. (btw, the stars aren't convolutions, just multiplication)

So then work out the next one in a jiffy.

$$[f(p),x]g=f*x*g-g*f*x=0$$

Well... according to my calculations, it is true that
$$\left[f(p),x\right] = i\hbar\frac{\mathrm{d}f}{\mathrm{d}p}$$
but not just because of symmetry. I'll second what Mindscrape wrote about the symbols not being just placeholders; they do have particular meanings.

What is the meaning of "true by symmetry" in a formula?

In mathematics, a formula is considered true by symmetry if it remains unchanged when certain elements or variables are interchanged. This means that the formula has a symmetric property, where the order of the elements does not affect its truth value.

What are the conditions for a formula to be considered true by symmetry?

In order for a formula to be true by symmetry, it must satisfy the following conditions:

• The formula must contain two or more elements or variables that can be interchanged.
• The formula must remain unchanged after the interchange of these elements.
• The interchange of elements must not change the truth value of the formula.

How can I identify if a formula is true by symmetry?

To identify if a formula is true by symmetry, you can perform a symmetry test by interchanging the elements or variables in the formula and checking if it remains unchanged. If the formula remains the same, it is considered true by symmetry.

Can a formula be true by symmetry for all types of mathematical operations?

Yes, a formula can be true by symmetry for all types of mathematical operations such as addition, subtraction, multiplication, and division. As long as the conditions for symmetry are satisfied, the formula can be considered true by symmetry.

Why is it important to understand symmetry in mathematical formulas?

Understanding symmetry in mathematical formulas can help identify patterns and relationships between different elements or variables. It also allows for the simplification and manipulation of complex formulas, making it easier to solve mathematical problems. Additionally, symmetry is a fundamental concept in mathematics and has applications in various fields such as physics, chemistry, and engineering.

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