Quantum mechanical expectation value

In summary, the quantum mechanical expectation value is the average result that one would expect from a measurement in a quantum mechanical system. It is related to uncertainty through the Heisenberg uncertainty principle and plays a fundamental role in quantum mechanics. It is calculated by taking the integral of a wave function multiplied by the corresponding operator and dividing by the total probability. While it cannot be measured directly, it can be calculated theoretically and used to validate the predictions of quantum mechanics.
  • #1
Master J
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I'm trying to calculate the expectation value of the momentum squared (p^2) of the harmonic oscillator ground state.

The integral involves the second derivative of a Gaussian (exponential of a negative squared term)

Then the integral involves, after working it out, an x^2 term times exp(-x^2).

I tried this by integrating by parts but it gets me no where. Am I missing something?
 
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  • #2
Wait, I solved it :p


If anyone is interestes, it's just the fact that the 2nd term is a derivative of a Gaussian!
 

1. What is the definition of quantum mechanical expectation value?

The quantum mechanical expectation value, also known as the expectation or average value, is the average result that one would expect from a measurement of a physical quantity in a quantum mechanical system. It is calculated by taking the average of all possible measurement outcomes, weighted by their corresponding probabilities.

2. How is the quantum mechanical expectation value related to uncertainty?

The quantum mechanical expectation value is related to uncertainty through the Heisenberg uncertainty principle. This principle states that the product of the uncertainty in position and the uncertainty in momentum of a quantum system cannot be smaller than a certain value. The expectation value of these quantities can be used to calculate the uncertainties, providing insight into the level of uncertainty in a system.

3. What is the significance of the quantum mechanical expectation value in quantum mechanics?

The quantum mechanical expectation value is a fundamental concept in quantum mechanics, providing a way to calculate and predict the behavior of quantum systems. It is used in various equations and principles, such as the Schrödinger equation and the Heisenberg uncertainty principle, and plays a crucial role in the understanding of quantum phenomena.

4. How is the quantum mechanical expectation value calculated?

The quantum mechanical expectation value is calculated by taking the integral of a wave function multiplied by the operator corresponding to the quantity being measured. This integral is then divided by the total probability of the system to obtain the average value. In simpler systems, the expectation value can be calculated using the sum of all possible measurement outcomes weighted by their probabilities.

5. Can the quantum mechanical expectation value be measured directly?

No, the quantum mechanical expectation value cannot be measured directly. This is because it represents an average value over all possible measurements, rather than a single measurement outcome. However, it can be calculated theoretically and compared to experimental results to validate the predictions of quantum mechanics.

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