How do I correctly evaluate <x^2> for a particle in a one-dimensional box?

In summary, the formula for the expectation value of x squared in an infinite one-dimensional box is (2/L)[(L^3/24)-(L^3/4n^2*pi^2)cos(n*pi)]. When n=1, the answer is (L^2/12)+(L^2/(2*pi^2)). However, the asker made a mistake when using the formula and arrived at (L^2/12)-(L^2/(8*pi^2)) instead. This mistake was corrected and the final answer was obtained.
  • #1
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Homework Statement



Consider a particle in an infinite one-dimensional box that has a length L and is centered at the origin. (Use h for Planck's constant, n, and L, as necessary.) Evaluate <x^2> for <x^2> at n=1.

Homework Equations



<x^2>= (2/L)[(L^3/24)-(L^3/4n^2*pi^2)cos(n*pi)]

The Attempt at a Solution



I used this formula and got the answer (L^2/12)+(L^2/(2*pi^2)). My assignment is telling me this is incorrect. I took the same approach for the second part of my homework asking for this when n=2, and I got (L^2/12)-(L^2/(8*pi^2)) which it says is correct. I cannot figure out why my first answer is not correct as well. Any help would be much appreciated.
 
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  • #2
Sorry. Just figured it out.
 

What is a quantum expectation value?

A quantum expectation value is a mathematical representation of the average value that a physical quantity will have in a quantum system. It is calculated by taking the sum of all possible outcomes of a measurement multiplied by their respective probabilities.

How is a quantum expectation value different from a classical expectation value?

In classical mechanics, the expectation value of a physical quantity is equivalent to the average value. However, in quantum mechanics, the expectation value is a probabilistic concept and does not necessarily correspond to the actual measurement value.

What is the significance of quantum expectation values in quantum mechanics?

Quantum expectation values are important in quantum mechanics because they allow us to make predictions about the behavior of quantum systems. They provide information about the potential outcomes of a measurement and the likelihood of each outcome occurring.

Can quantum expectation values be measured?

No, quantum expectation values cannot be measured directly. They are purely theoretical constructs that represent the average value of a physical quantity in a quantum system. However, they can be calculated and compared to experimental results to validate the predictions of quantum mechanics.

How can quantum expectation values be used in practical applications?

Quantum expectation values are used in practical applications such as quantum computing, where they are used to calculate the expected outcomes of operations on quantum bits (qubits). They are also used in quantum chemistry to predict the behavior of atoms and molecules, and in quantum cryptography for secure communication.

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