Calculate the commutator [p x , x[SUP]n[/SUP]]

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In summary, the commutator [p x, x<SUP>n</SUP>] is a mathematical operation used in quantum mechanics to calculate the uncertainty between the momentum operator (p) and the position operator (x<SUP>n</SUP>). It is significant because it helps us understand and quantify the uncertainty principle, and it is related to Heisenberg's uncertainty principle. To calculate the commutator, you must expand the expression using the commutator rule and substitute the momentum and position operators. It can never be equal to zero due to the non-commutative nature of the operators.
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hanspl
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Calculate the commutator [px, xn]

Homework Statement



There are 3 tasks.
1) No other information is given. only that I have to calculate the commutator [px, xn].

For task 2 and 3 a relevant equation is given below.

2) calculate the commutator [x, Kx]

3) calculate the commutator [px, Kx]

Homework Equations



We define the kinetic energy operator of a particle in one dimension (x) as

Kx = - (h(hatch)2/2m)(d2/dx2)

I hope anyone can help me. I am really stuck with these commutators.

- HP
 
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  • #2
Try multiplying the commutator at the right by some phantom ket |K>...
 

1. What is the commutator [p x, xn]?

The commutator [p x, xn] is a mathematical operation used in quantum mechanics to calculate the uncertainty between the momentum operator (p) and the position operator (xn). It is used to determine the likelihood of a particle's position and momentum being accurately measured at the same time.

2. How do you calculate the commutator [p x, xn]?

To calculate the commutator [p x, xn], you first need to expand the expression using the commutator rule [A,BC] = [A,B]C + B[A,C]. Then, you can substitute the momentum operator (p) and the position operator (xn) into the equation and simplify to get the final result.

3. What is the significance of the commutator [p x, xn] in quantum mechanics?

The commutator [p x, xn] is significant in quantum mechanics because it helps us understand the uncertainty principle, which states that the more precisely we measure the position of a particle, the less precisely we can measure its momentum, and vice versa. This commutator allows us to quantitatively calculate this uncertainty.

4. Can the commutator [p x, xn] ever be equal to zero?

No, the commutator [p x, xn] can never be equal to zero. This is because the position and momentum operators do not commute, meaning that their order in an equation matters and switching their positions will result in a different outcome. Therefore, the commutator will always have a non-zero value.

5. How is the commutator [p x, xn] related to Heisenberg's uncertainty principle?

The commutator [p x, xn] is directly related to Heisenberg's uncertainty principle. It is used to calculate the uncertainty between the position and momentum of a particle, which is a fundamental principle of quantum mechanics. The commutator helps us understand and quantify this uncertainty, providing a mathematical foundation for Heisenberg's uncertainty principle.

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