# One dimensional Quantum Mechanics problems

1,

Show that for any normalized |u> , the inequality E0 <= <u|H|u> holds, where E0 is teh lowest energy eigenvalue.

Prove the following theorem: Every attractive potential in one dimension has at least one bound state.

2.

Consider the potential V = -aV0 * delta(x)

Show that it admits a bound state of energy E = -ma^2 * V0^2 / 2h^2

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#### Solution Preview

Please see the attachment.

Let the eigenbasis of the Hamiltonian H be denoted by such that:

(1.1)

Also we can order the energy levels in such a way that:

(1.2)

Now, we can write the normalized wave function as a linear combination of the eigenstates:

(1.3)

And since is normalized we have:

(1.4)

Now,

(1.5)

Due to (1.2) and the fact that we can write equation (1.5) as an inequality:

(1.6)

Using (1.4) ...

#### Solution Summary

The 8 pages solution shows in detail how to approach and solve the problems.