Computing Heisenberg Uncertainty Value

In summary, the problem involves a particle in a one dimensional box of length L and a potential energy function. The task is to compute ΔxΔp, using the equations Δx and Δp, and given the wave function at ground state ψ=sqrt(2/L)sin (pi*x/L). The solution requires evaluating the integral from 0 to L x^2sin^2(pi*x/L)dx, which can be simplified using a trig identity and integration by parts.
  • #1
i.nagi
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Homework Statement



Consider a particle in a one dimensional box of length L, whose potential energy is V(x)=0 for 0<x<L, and infinite otherwise.Given the wave function at ground state ψ=sqrt(2/L)sin (pi*x/L) Compute ΔxΔp where

Homework Equations



Δx=sqrt(<x^2>-<x>^2) and Δp=sqrt(<p^2>-<p>^2)

The Attempt at a Solution


I have set the expected value for as <x^2> equal to the integral from 0 to L ψ1(x)(x^2)ψ1(x)dx, as done by my prof. I then evaluated this to 2/L*the integral from 0 to L x^2sin^2(pi*x/L)dx. However I am stuck here, and my prof's solution goes straight to L^2(1/3-1/2pi^2). I am confused as to how he evaluated this integral and the role of (x^2) in the equation, if steps could be shown that would help immensely, also when calculating <x>, is x included in the integral, and how might it be evaluated if included with sin^2(pi*x/L). Thanks in advance.
 
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  • #2
This is just integration not QM. But you first step is to use a trig identity sin(u)^2=(1-cos(2u))/2 to get rid of the power on the sin. Now multiply it out and start using integration by parts to deal with the x^2*cos part. You must have seen these things somewhere before.
 

What is the Heisenberg Uncertainty Principle?

The Heisenberg Uncertainty Principle is a fundamental principle in quantum mechanics that states that it is impossible to know both the exact position and momentum of a particle at the same time. This means that the more accurately we measure one of these properties, the less accurately we can measure the other.

What is the significance of computing Heisenberg Uncertainty Value?

Computing the Heisenberg Uncertainty Value allows us to determine the minimum uncertainty in the measurement of a particle's position and momentum. This is important in understanding the limitations and uncertainties in quantum mechanics and can also be useful in practical applications such as quantum computing.

How is the Heisenberg Uncertainty Value calculated?

The Heisenberg Uncertainty Value is calculated by taking the product of the uncertainties in a particle's position and momentum. This value is equal to or greater than Planck's constant divided by 2π.

Can the Heisenberg Uncertainty Value be overcome?

No, the Heisenberg Uncertainty Value is a fundamental principle in quantum mechanics and cannot be overcome. However, there are ways to reduce the uncertainty in one property by sacrificing accuracy in the other, such as in the case of entangled particles.

How does the Heisenberg Uncertainty Value relate to the wave-particle duality?

The Heisenberg Uncertainty Value is a result of the wave-particle duality of quantum particles. This duality states that particles can exhibit both wave-like and particle-like behaviors, making it impossible to know both their position and momentum with certainty at the same time.

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