Bound states

In quantum mechanics, bound states refer to the specific energy states of a quantum system that are confined or "bound" within a potential energy well. These bound states are characterized by the fact that the energy of the particle is below a certain threshold, and the particle is localized within a limited region of space due to the potential energy barrier surrounding it.

Bound states are in contrast to unbound or scattering states, where the energy of the particle is above the threshold, and the particle can move freely through space, interacting with the potential but not being trapped within it.

The states of a system or of a particle which is localized a region are known as bound states. The particle confined in a region can not escape to infinity. So wavefunction of the particle in bound state must tend to zero at infinity. Wavefunction describing bound state are square integrate.

When the potential energy of a particle has minimum value Vo, the particle is said to be in potential well. We are going to show that total energy of the particles E in bound state is greater than - Vo i.e E> - Vo

K.E operator

Expectation value of T = < T ><u(x)>l T l u(x)>

Expectation value of potential energy operator

< V> = ∫ u*(x)v(x)u(x) dX

Again

We have Hu(x) = E u(x)

Multiplying throughout by u*(x) and then integrating

<T> + <V> = E

It is obvious that < V> ⩾ - Vo

Since <T> > 0

And <V> ⩾ - Vo

We can conclude that E > - Vo

In classical physics when the particle is at rest at the position where V = - Vo is minimum

< T > = 0

E = - Vo

Quantum mechanically in bound state the particle is confined in the region. If the size of the region is ΔX according to uncertainity principle. ΔX. ΔPx ⩾ℏ. ΔPx has certain values then

ΔPx ≈ ℏ/ ΔX

This indicates that the particle has certain momentum and it ahs kinetic energy so that particle cannot be at rest. So E>-Vo. We are going to show that when the particle is in bound state, in one dimensional case, the energy eigen value is non degenerate and the energy eigen function are real.

Let us purpose that there are two linearly independent normalized eigen functions u_n(x) and u_m(x) belonging to degenerate eigen values

En = Em

Multiplying 1 by u_m(x) and 2 by u_n(x) and subtract 2 from 1

Wronsksian of the function u_m(x) and 2 by u_n(x)

Since wavefunction vanish at infinity

u_m(X) -> 0 at X = ± ∞

u_n(X) -> 0 at X = ± ∞

So W = 0 at X = ± ∞

Since W = constant

W =0 everywhere in the whole region

Since both u_n(x) and u_m(x) are normalized

C = 1

So u_n(x) = u_m(x)

So there is only one eigen function of H with eigen value E. So E is non degenerate.

If we repeat the whole process mentioned about taking u*_n(x) in place of u_m(x) it will be seen that

u_n(x) = u*_n(x)

This indicates that u_n(x) is real.

In more complex systems, such as atoms, molecules, and solid-state materials, bound states also play a crucial role. In an atom, for example, the electron is bound to the nucleus due to the Coulomb potential. The electron's energy levels are quantized, and each energy level corresponds to a different bound state around the nucleus.

Bound states have significant implications in various physical phenomena, including the stability of matter, the behavior of electrons in crystals (band structure), the quantization of energy levels in atomic and molecular systems, and the formation of chemical bonds.

It's worth noting that bound states are a fundamental aspect of quantum mechanics, and they are a consequence of the wave-like nature of particles and the quantization of energy levels in quantum systems.

This note is a part of the Physics Repository.