Estimating Ground State Energy Using Heisenberg's Uncertainty Principle

[ee7klt]

hi,
if we have a potential of the form V(x) = a|x|, how would one go about estimating the ground state energy of the system using the Heisenberg uncertainty principle.

I suppose the thing to do is to get an estimate of delta x, then get delta p via the Heisenberg uncertainty principle and the E = delta p^2/2m but the particle could be anywhere on the real axis since the potential is finite on the interval (-inf,inf)!
any hint much appreciated

 

[gouthambs]

I don't understand how you would estimate the ground state energy using Heisenberg uncertainity principle. However what I would do to estimate the groundstate energy is to use the variational principle. To use the variational principle you need to guess a form of the ground state wave function. So for this case the ground state wave function would be symmetric. So try a Gaussian form psi(x) = exp(-x^2/a^2), with a being the variational parameter. You can refer to "Modern Quantum Mechanics", JJ Sakurai, pg 313.
Goutham

 

[R0man]

Estimating the ground state energy using Heisenberg's uncertainty principle is a commonly used method in quantum mechanics. In order to estimate the ground state energy of a system with a potential of the form V(x) = a|x|, we first need to determine the uncertainty in position, delta x. This can be done by considering the potential and finding the region where the probability of finding the particle is highest. In this case, it would be in the region near x=0. Therefore, we can estimate delta x to be the width of this region.

Next, we can use the Heisenberg uncertainty principle, which states that the product of the uncertainties in position and momentum is always greater than or equal to h/4π, where h is Planck's constant. So, we can estimate the uncertainty in momentum, delta p, to be h/4π*delta x.

Finally, we can use the equation E = delta p^2/2m to estimate the ground state energy. Plugging in our estimated values for delta p and solving for E, we can get an approximate value for the ground state energy.

However, as you have mentioned, the particle could be anywhere on the real axis since the potential is finite on the interval (-inf,inf). This is where the uncertainty principle comes into play. It tells us that the particle's position and momentum cannot be known simultaneously with absolute certainty. So, while our estimation may not give us the exact ground state energy, it gives us a reasonable approximation based on the principles of quantum mechanics.

In summary, to estimate the ground state energy using the Heisenberg uncertainty principle, we need to first estimate the uncertainty in position, then use the uncertainty principle to estimate the uncertainty in momentum, and finally use the equation E = delta p^2/2m to estimate the ground state energy. While this may not give us the exact value, it provides a useful estimate for understanding the behavior of the system.