Which Potential Has a Lower Ground State Energy?

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SUMMARY

The discussion centers on comparing the ground state energies of two one-dimensional potentials: V0(x) and V1(x). The first potential, V0(x), is defined as zero within the interval abs|x| ≤ a/2 and infinite outside, while the second potential, V1(x), is a linear potential defined as (4V|x|)/a - V_0 within the same interval. The conclusion drawn is that E0, the ground state energy of V0, is less than E1, the ground state energy of V1, due to the fixed nature of E0 compared to the variable nature of E1 influenced by V0.

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  • Understanding of quantum mechanics, specifically the Schrödinger Equation.
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  • Explore the mathematical solutions to the Time-Independent Schrödinger Equation (T.I.S.E.) for various potential forms.
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Students and researchers in quantum mechanics, particularly those focusing on potential energy problems and ground state energy comparisons in one-dimensional systems.

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Homework Statement



Let ϕn(x) be the complete ortho-normal set of eigenstates of the Hamiltonian H = T +V
and En, n = 0, 1, 2, are the corresponding eigenvalues. The E0 is the ground state energy.
(a) If ϕ(x) is an arbitrary normalized state, show that E0 < = ∫dxϕ(x)Hϕ(x).

(b) Consider one dimensional potential V0(x) = 0, abs|x| ≤ a/2 and = ∞ otherwise. The
second one dimensional potential V1(x) = (4V|x|)/a - V_0, |x|≤ a/2, V > 0 and = ∞otherwise.
If E0 and E1 are the ground state energies of the two potentials respectively. Find which
one (1) E0 < E1 (2) E0 > E1 is correct and prove your judgement.

Homework Equations



Schrödinger's Equation

The Attempt at a Solution



I think I have gotten part A correct. However, I am so confused at a non-square well potential. I know that the infinite potential boundary conditions still apply here, but it throws me off with what is going on inside the well.

Do I just do a normal solution of the T.I.S.E. and let my constant take care of the non-zero potential?

Any help is greatly appreciated.
 
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Wouldn't the answer depend on the value of V0? You can shift the value of E1 around by changing V0, but E0 is fixed.
 

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