Study Notes on Quantum Mechanics in Three Dimensions and Hydrogen Atom

HELSINGIN YLIOPISTO

UNIVERSITY OF HELSINKI

Quantum Mechanics in Three Dimensions and H atom

Faculty of Science

19th January 2026

1. SCHRÖDINGER EQUATION IN 3D

1.1 Introduction to the Schrödinger Equation

• The Schrödinger equation in three dimensions (3D) is an extension of the one-dimensional (1D) case.

• In transitioning from 1D to 3D, the equation incorporates the spatial coordinates represented as
  $extbf{r} = (x, y, z)$

• To eliminate coordinate bias, the Schrödinger equation is often expressed in a coordinate-independent form.

1.2 Kinetic Energy and Laplacian

• The kinetic energy term in the equation can be replaced with the Laplacian operator, which is a fundamental operator in calculus utilized in physics to describe various phenomena.

2. PROBABILITY DENSITY AND NORMALIZATION IN 3D

2.1 Wave Function and Probability

• In one dimension, the complex square of the wave function represents the probability per unit length.

• For three dimensions, it expresses the probability per unit volume where the probability density function is given by
  $|Ψ( extbf{r}, t)|^2$ .

• Normalization condition for a single particle in 3D requires:
  $ext{Probability} = <br>ightpadding{ ext{Total Probability}} = rac{1}{V} = 1$

• The dimensions of the wave function in three dimensions are determined as:
  $[ ext{Wave Function}] = L^{-3/2}$

3. TIME-DEPENDENT SCHRÖDINGER EQUATION

3.1 Temporal and Spatial Separation

• We consider states where energy remains constant over time, assuming the wave function can be decomposed into spatial and temporal components:
  $Ψ( extbf{r}, t) = φ(t)χ( extbf{r})$

3.2 Deriving the Equation

1. Substitute this product into the standard 3D Schrödinger equation.

2. Divide the equation by the wave function to express in separate temporal and spatial forms.

3. The right side becomes a function solely dependent on time and the left side only on position.

3.3 Constants and Implications

• This necessitates both sides being constant, represented as some constant value C.

• Therefore, temporal part (
  $φ(t)$) and spatial part (
  $χ( extbf{r})$) can be separated.

4. TIME-INDEPENDENT SCHRÖDINGER EQUATION

4.1 Equation Structure

• The time-independent Schrödinger equation can be derived when the wave function derivative equals the energy:
  $- rac{ℏ^2}{2m} <br>abla^2 χ( extbf{r}) + U( extbf{r})χ( extbf{r}) = E χ( extbf{r})$

• The potential energy function U is critical in determining the specific form of the equation.

5. QUANTUM DOT: THE 3D INFINITE WELL

5.1 Basic Concept

• The simplest three-dimensional bounded system is modeled by the 3D infinite well or box, a relevant representation of quantum dots.

• The particle is constrained within a box-shaped region defined by infinitely high potential barriers at the coordinates:

  • x = 0 and Lx

  • y = 0 and Ly

  • z = 0 and Lz

5.2 Potential Energy Function

• The potential energy is denoted as follows:
  U( extbf{r}) = egin{cases} 0 & ext{for } 0 < x < Lx, 0 < y < Ly, 0 < z < Lz \ \ ext{∞} & ext{otherwise} \ \ ext{for } ext{3D Box} ext{ with dimensions } Lx, Ly, Lz \ ext{where the particle is confined in the mentioned dimensions.} \ \ ext{Thus the potential within the region is zero while}\ ext{it is infinite outside,} \ ext{effectively trapping the particle.} \ ext{Explained further by defining the wave functions for each spatial variable.} \ \ extbf{Separation of Variables}[PS] \ - The spatial wave function can be expressed as a product of three independent functions corresponding to each variable:
  Ψ(x, y, z) = F(x)G(y)H(z) $$