2.3 Intro

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2.3.1 Wavefunction and its interpretations

I Wavefunction($\psi$): A mathematical function with values that vary with position.

1. replacement of precise trajectories of a particle to account for wave-particle duality.

2. Born interpretation: probability of finding a particle in a specific region is proportional to the value of $\psi$ in that region.

3. Probability density ($\psi^{2}$): the probability of a particle to be found in a small region divided by the volume of the region.

   1. analogous to mass density: mass divided by volume

4. node: where the wavefunction passes through a value of zero

   1. A position where there is zero probability of an electron to exist

   2. there are some debates on whether or not a node actually has zero probability

II. Schrödinger equation: A mathematical equation used to calculate both the wavefunction and its corresponding energy.

III. One of its solutions, referred to as time-independent Schrödinger equation (TISE), for a particle moving in one dimension is: $- \frac{-\hbar^{2}}{2m} \frac{∂^{2}\psi}{∂x^{2}} + V(x)\psi = E\psi$

1. The terms $- \frac{-\hbar^{2}}{2m} \frac{∂^{2}\psi}{∂x^{2}}$ represent kinetic energy, and V(x)ψrepresent potential energy. Together they are often referred to as the Hamiltonian of the system, written as $H\phi$.

2. The total energy of a particle (E) equals the kinetic energy of the particle (T) and the potential energy of the particle (V); E = T + V.

3. Because $T_{k} = \frac{1}{2}mv^{2}$, this can also be $T_{k} = \frac{p^{2}}{2m}$. Using Einstein's light quanta thesis (E = h\nu = ħ\omega), De Brogile's equation (Eq. 7), and some linear algebra, the kinetic energy term $- \frac{-\hbar^{2}}{2m} \frac{∂^{2}\psi}{∂x^{2}}$ can be derived.

4. the term $H\psi$ can be thought of as the total energy of the wavefunction $\psi$.

IV. The Schrodinger equation can be simplified further using the particle-in-a-box model.

2.3.2 The Quantization of Energy

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V. A particle in a box: a simple one-dimensional box that confines a single particle with mass of m

1. The particle exhibits wave-particle duality

2. only certain wavelengths can exist in the box; must be half-periods

3. The walls of the box has infinite potential energy, while inside it is zero

4. The wavefunction for this particle-in-a-box is:

   $\psi\_{n}(x) = (\frac{2}{L^{\frac{1}{2}}} sin(\frac{n \pi x}{L}))$ (Eq. 9)

VI. quantum number: an integer or half-integer that labels a wavefunction that specifies a state or a property.

1. The number n in a particle-in-a-box is an integer that labels the wavefunctions; higher n represents higher state of energy.

VII. Because n can only take on integer values, we can say that energy is quantized: energy is restricted to a series of specific values called energy levels.

1. This applies to the three-dimensional world as well

2. Note that the quantum mechanical description of energy has a defined "smallest amount of energy that can be transferred" but classical mechanics does not.

3. An atom has similar quantized energy levels to a particle-in-a-box

   1. its wavefunction has specific boundary constraints (conditions) similar to the particle-in-a-box model

   2. due to these constraints, only a few solutions of its Schrödinger equation and its corresponding energies exist.

VIII. The equation $E_{k} = \frac{n^{2}h^{2}}{8mL^{2}}$ can be used to calculate the energy separation between two neighboring levels with quantum numbers n and n + 1 using the equation:

$E_{n+1} - E_{n} = {(n + 1)^{2} - n^{2}} \frac{h^{2}}{8mL^{2}} = (2n + 1)\frac{h^{2}}{8mL^{2}}$ (Eq. 11)

1. As L (the length of the box) increases, the separation between neighboring energy level decreases

2. As m (the pass of the particle) increases, the separation between neighboring energy level decreases

3. As the lowest possible quantum number (n) is 1, the lowest possible energy is known as the zero-point energy and is $E_{k} = \frac{h^{2}}{8mL^{2}} ≈ 10^{-67}J$ (A particle cannot be perfectly still.)

Why is this important?

1. The energy levels of the hydrogen atom are explained by this property. As you go up to higher energy levels in a Bohr model, the separation of energy levels decreases.

2. In macroscopic environments, quantization is completely undetectable and is negligible; hence why classical mechanics work in most cases.

3. The concept of zero-point energy is consistent with the uncertainty principle. A particle cannot possess zero energy as that would make the uncertainty of both the momentum and position of a particle also equal to zero, as it is not moving. This scenario defies the uncertainty principle and cannot happen.