*Exploring Quantum Physics – Week 3, Lecture 5*

## Types of problems for the Schrödinger equation

Single-particle quantum mechanics,

## Quantization in a guitar string and quantum well

- Wavelength “quantization” in a guitar string: String anchored at end nodes, must have multiples of a half-wave between them

- Wavelength quantization in an infinite quantum well: potential “walls” anchor the wave function similar to guitar string

## Formulation of the finite potential well problem

- For
- For
- Find the bound state(s) (with ) satisfying continuity constraints (problem definition):

*Where +0 means to the right of a point, and -0 means to the left*

**Using the symmetry**

- If the Hamiltonian commutes with an operator, , then solutions to the Schrödinger equation can be chosen to have definite
*a*and*E*,

- Potential is then inversion symmetric, . Eigenvalues of are
- General solution of :

- Can choose solutions with definite parity, i.e.

**Using the constraints at infinities**

- General solution of :

- E.g. for
*x > a/2*we must request that as . Otherwise probability would not be finite (does not make sense). - So drop
*B*term and the solution is

**Using matching conditions at transition points**

- $latex \displaystyle \psi_+(x) = \left\{ \begin{array}{ll}

A e^{\gamma x} & x < -a/2 \\

C \cos(kx) & |x| a/2

\end{array} \right. $ - Match solutions at
*x = a/2*

Divide Eq. 2 / Eq. 1:

**Making the self-consistency equation dimensionless**

- The non-linear self-consistency equation is not solvable analytically:

with - Start by introducing dimensionless parameters, and

- 2 limiting cases:
- Deep well:
- Shallow well:

**Solving**

## Bound status in quantum potential wells

**Fourier transforms**

- Can solve for shallow potential well, using Fourier transform:

- Specifically for Dirac -function:

- Use these identities to solve stationary Schrödinger equation for (with )

**Shallow level in the delta-well**

*What is E?*

Take the integral of :

…

**Final result:**

## Increasing to *D* dimensions

In 2 dimensions: , where is the angle of the k-vector

In D dimensions:

*where from the integral over the angle *

**Critical dimension (=2)**

- Main question: is there a bound state in a weak potential ?
- Can we make arbitrarily large by choosing
*E*?- Equivalent to asking whether diverges
- For D=2,

- If , it diverges (there is a bound state), otherwise it is finite (no bound state).

**Very shallow level in 2D quantum well**

**Result:**

As , this result has no Taylor expansion.

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