Quantum Mechanics Notes

Probability Calculations and Eigenfunctions

• The advantage of expressing the wave function in terms of eigenfunctions is to calculate probabilities.
  • For discrete eigenvalues, the square of the coefficients represents the probabilities.
  • For continuous eigenvalues, the square of the inner product provides the probability density.

Inner Product and Coefficient Determination

• Inner product is used to find the components of a wavefunction.

• The x component is obtained via the inner product:

  • Analogy: If $\vec{r} = a \hat{x} + b \hat{y}$, then $a = \vec{r} \cdot \hat{x}$.
  • Only the term corresponding to the eigenfunction survives when the dot product is performed.

• The inner product allows calculation of coefficients:

  • If $\psi(x, t=0)$ is known, the coefficients can be determined.
  • The square of these coefficients gives the probability of measuring an observable.

Inner Product Details

• Inner product is analogous to a dot product. It involves a complex conjugate, important for imaginary eigenfunctions.
• Example problem: A particle is confined between $x = 0$ and $x = L$.
  • Part a: Find $b<em>k$, the coefficient on the momentum eigenfunction. $b</em>k^2$ corresponds to the probability density in k-space.
  • Part b: Find the coefficient $c$ for $x = L/2$. $c^2$ corresponds to the probability density in real space (position eigenfunctions).

Delta Function and Probability Density

• Using a delta function, with height $1/\sqrt{a}$ and width $a$, to calculate the inner product.
• The probability of finding the particle between $L/2$ and $L/2 + a$ is to be determined.
• Both momentum and position operators have continuous eigenvalues leading to probability densities.

Integral Forms and Evaluation

• For the momentum operator case, the answer can be left in integral form.
• For the position operator case, the integral should be evaluated.
• To find $b_k$, the complex conjugate of the momentum eigenfunction is needed:
  • $e^{ikx}$
  • The function is $\psi(x) = e^{-x/L}$.
  • $b<em>k = \int</em>{-\infty}^{\infty} e^{-ikx} \psi(x) dx = \int_{-\infty}^{\infty} e^{-ikx} e^{-x/L} dx$
  • $b<em>k$ will be an imaginary number, but $|b</em>k|^2$ will be real, as required for probability density.

Fourier Transform Connection

• The calculation of $b_k$ resembles a Fourier transform of the wave function.

Probability Calculation with Delta Function

• Integrating from $-\infty$ to $\infty$, but the delta function is only non-zero at $x = L/2$.

• The integral becomes the wave function evaluated at $L/2$.

• $\int_{-\infty}^{\infty} \delta(x - L/2) \psi(x) dx = \psi(L/2)$

• With the delta function: height $1/\sqrt{a}$, width $a$

• The result of the integral is $\psi(L/2) / \sqrt{a}$.

• Squaring this gives the probability density: $\left| \frac{\psi(L/2)}{\sqrt{a}} \right|^2$

• This $c^2$ represents a probability from $L/2$ to $L/2 + a$, not a probability density because we're not working in the limit of an infinitesimally small $dx$.

Generalization to Other Operators

• This procedure applies to any operator.
• For finding the probability of energy between $E$ and $E + \delta E$, use the Hamiltonian eigenfunction.
• Example: Kinetic energy eigenfunctions for energy range (e.g., 1 joule to $1.3 \times 10^{-17}$ joules).

Practical Measurement and Wave Function Role

• Practical measurement involves detectors. Detector size can correspond to 'a'.
• The wave function is analogous to position and velocity in classical mechanics.
• From the wave function, kinetic energy, momentum, and probabilities can be calculated.

Hamiltonian Operator

• The Hamiltonian operator $H$ is the sum of kinetic energy operator $T$ and potential energy operator $V$: $H = T + V$.
• $H$ operates on the spatial part of the wave function.

Potential Energy Operator

• Potential energy $V$ is a function of position $x$.
• Example: $V(x) = \frac{1}{2} kx^2$ (harmonic oscillator potential).
• This could model an electron acted upon by a spring-like force.

Potential Energy and Systems

• Potential energy always relates to a system (e.g., electron-proton system in the hydrogen atom).
• For the hydrogen atom, $V(r) = -\frac{ke^2}{r}$, where $r$ is the distance between proton and electron.
• The potential energy is a function of position and converted into an operator.

Forming the Potential Energy Operator

• Replace position variables with operators.
• For $V(x) = \frac{1}{2} kx^2$, the operator is $\hat{V} = \frac{1}{2} k \hat{x} \hat{x}$.
• Applying operator twice, not squaring it. Not $\hat{x}^2$ but $\hat{x}(\hat{x} f(x))$.

Uniqueness of Quantum Mechanical Problems

• Each quantum mechanics problem is unique due to its potential function $V(x)$.
• Different $V(x)$ changes the Hamiltonian, thus changing the energy eigenstates.
• Kinetic energy operator is the same for different interactions between the particle and surroundings.

Classical vs. Quantum Mechanics

• Classical Mechanics: Problems differ due to forces acting on particle.
• Quantum Mechanics: Problems differ due to potential energy function.
• Newton's second law: $F = ma$.
• In quantum mechanics, knowing the potential energy function is essential. No potential energy means the problem is a free particle.

Free Particle

• If the potential energy $V = 0$, it's called a free particle problem.
• The Hamiltonian is only the kinetic energy operator.
• Potential energies can be derived from forces: $\Delta V = -\int<em>{x</em>1}^{x_2} F dx$

Connecting Classical and Quantum Mechanics

• Classical force can be used to find potential, which then becomes an operator for the Hamiltonian.
• This applies at the atomic level where quantum mechanics is used.
• Semiclassical models like Bohr's use classical mechanics to derive quantum models.

Energy Eigenfunctions and Potentials

• Energy eigenfunctions depend on the potential energy.
• To find eigenfunctions, solve the eigenvalue problem for the Hamiltonian.

Particle in a Box Problem

• Simplest quantum mechanics problem.

• Free particle eigenfunctions are also Hamiltonian eigenfunctions since $H = T$.

• Box potential: Potential energy is written as $V$ instead of $PE$ (common for electron problems, where $V$ stands for voltage).

• Potential: $V(x) = 0$ for 0 < x < L, $V(x) = V_0$ elsewhere.

Classical Understanding of the Box

• Classically, the particle is trapped in the box because the force is large at the boundaries.
• Take the limit as V0 approaches infinity for trapping.

Quantum Mechanical Treatment

• Time-independent Schrodinger equation $H \psi = E \psi$.
• In regions where $V$ approaches infinity, $\psi$ must approach zero.
• $\psi(x) = 0$ for $x < 0$ and $x > L$.
• Both classical and quantum mechanics agree on this point.

Solving Inside the Box

• Inside the box (0 < x < L), $V = 0$, so $H = T$.
• Schrodinger equation becomes: $-\frac{\hbar^2}{2m} \frac{d^2 \psi}{dx^2} = E \psi$.
• The function must be continuous and differentiable, so $\psi(0) = 0$ and $\psi(L) = 0$.

Boundary Conditions

• Since $\psi$ has to be zero at $x=0$ and $x=L$.
• General solution is $\psi(x) = A \sin(kx)$, where $k = \sqrt{\frac{2mE}{\hbar^2}}$.
• To satisfy the condition $\psi(L) = 0$, $kL = n\pi$, where $n$ is an integer.

Quantization of Energy

• Energy is quantized: $E_n = \frac{n^2 \pi^2 \hbar^2}{2mL^2}$, where n is an integer.
• The cosine function is not suitable because $\cos(0) = 1 \neq 0$.
• Sine function and linear combinations of exponential functions work because they vanish at $x = 0$.

Discrete Energy Values and Eigenfunctions

• Energy eigenfunctions are discrete and can be denoted by $\psi_n(x)$.
• The solutions are of the form $\psi_n(x) = A \sin(\frac{n \pi x}{L})$ where $n = 1, 2, 3…$
• Examples of Wavefunctions:
  • n = 1; A single hump
  • n = 2; two humps

Probability Density

• Squaring the wave function gives probability density $|\psi_n(x)|^2$.
• For $n = 2$, the probability density is zero at the middle of the box $\psi(x=L/2)$.
• In Classical Mechanics the probability is uniform across the box.
• QM is different, there are positions where you will not measure the particle.

Space Quantization

• Quantum mechanics results in space quantization, where particles are more likely to be found in certain locations.
• This is analogous to Bohr's model where electrons can only be in certain orbits.

Large n Values and Correspondence Principle

• For large n, the quantum mechanical probability density tends towards the classical prediction of uniform distribution.
• This aligns with Bohr's correspondence principle.