IMG_2997.jpeg

# Quantum Mechanics ## What is Quantum Mechanics? Quantum mechanics is a branch of physics that deals with the behavior of matter and energy at the atomic and subatomic levels. It is based on the idea that energy is quantized, meaning that it can only exist in discrete amounts. This is in contrast...

# Quantum Mechanics ## What is Quantum Mechanics? Quantum mechanics is a branch of physics that deals with the behavior of matter and energy at the atomic and subatomic levels. It is based on the idea that energy is quantized, meaning that it can only exist in discrete amounts. This is in contrast to classical physics, which assumes that energy can take on any continuous value. ### Key Concepts * **Quantization:** Energy, momentum, and other physical quantities are quantized, meaning they can only exist in discrete values. * **Wave-particle duality:** Particles can behave as waves, and waves can behave as particles. This is demonstrated by the famous double-slit experiment. * **Uncertainty principle:** It is impossible to know both the position and momentum of a particle with perfect accuracy. The more accurately we know one, the less accurately we know the other. Mathematically, this is expressed as: $$\Delta x \Delta p \ge \frac{\hbar}{2}$$ where: * $\Delta x$ is the uncertainty in position * $\Delta p$ is the uncertainty in momentum * $\hbar$ is the reduced Planck constant $(\hbar = h / 2\pi)$ * **Superposition:** A particle can be in multiple states at the same time. For example, an electron can be in multiple energy levels simultaneously. * **Entanglement:** Two or more particles can be linked together in such a way that they share the same fate, no matter how far apart they are. ### Mathematical Framework The state of a quantum system is described by a wave function, denoted by $\Psi$. The wave function contains all the information that can be known about the system. The wave function evolves in time according to the Schrödinger equation: $$i\hbar \frac{\partial \Psi}{\partial t} = \hat{H} \Psi$$ where: * $i$ is the imaginary unit * $t$ is time * $\hat{H}$ is the Hamiltonian operator, which represents the total energy of the system. ### Applications Quantum mechanics has many applications in modern technology, including: * **Lasers:** Lasers rely on the principle of stimulated emission, which is a quantum mechanical phenomenon. * **Transistors:** Transistors are the building blocks of modern computers and other electronic devices. They rely on the quantum mechanical properties of semiconductors. * **Magnetic Resonance Imaging (MRI):** MRI is a medical imaging technique that uses the quantum mechanical properties of atomic nuclei to create images of the inside of the body. * **Quantum Computing:** Quantum computers are a new type of computer that uses quantum mechanical phenomena to perform calculations that are impossible for classical computers. ### Further Study * **Griffiths, David J. "Introduction to Quantum Mechanics."** * **Shankar, R. "Principles of Quantum Mechanics."** ## Problems 1. A particle is confined to a one-dimensional box of length $L$. The potential energy is zero inside the box and infinite outside the box. (a) Write the time-independent Schrödinger equation for this system. (b) Solve the Schrödinger equation to find the allowed energy levels and corresponding wave functions. (c) Calculate the probability of finding the particle in the region $0 < x < L/2$ for the ground state. 2. Consider a harmonic oscillator with potential energy $V(x) = \frac{1}{2}m\omega^2 x^2$. (a) Write the time-independent Schrödinger equation for this system. (b) Find the energy levels of the harmonic oscillator. (c) Determine the wave function for the ground state. 3. An electron is in the spin state: $$|\chi\rangle = A \begin{pmatrix} 1+i \\ 1 \end{pmatrix}$$ (a) Find the normalization constant $A$. (b) What are the probabilities of measuring $S_z = \hbar/2$ and $S_z = -\hbar/2$? (c) Calculate the expectation value of $S_x$. ## Solutions ### Problem 1 (a) The time-independent Schrödinger equation for a particle in a 1D box is: $$-\frac{\hbar^2}{2m} \frac{d^2 \psi(x)}{dx^2} = E \psi(x)$$ (b) Solving the Schrödinger equation gives the allowed energy levels: $$E_n = \frac{n^2 \pi^2 \hbar^2}{2mL^2}, \quad n = 1, 2, 3, \dots$$ and the corresponding wave functions: $$\psi_n(x) = \sqrt{\frac{2}{L}} \sin\left(\frac{n\pi x}{L}\right)$$ (c) For the ground state ($n=1$), the probability of finding the particle in the region $0 < x < L/2$ is: $$P = \int_0^{L/2} |\psi_1(x)|^2 dx = \int_0^{L/2} \frac{2}{L} \sin^2\left(\frac{\pi x}{L}\right) dx = \frac{1}{2}$$ ### Problem 2 (a) The time-independent Schrödinger equation for a harmonic oscillator is: $$-\frac{\hbar^2}{2m} \frac{d^2 \psi(x)}{dx^2} + \frac{1}{2}m\omega^2 x^2 \psi(x) = E \psi(x)$$ (b) The energy levels of the harmonic oscillator are: $$E_n = \hbar\omega\left(n + \frac{1}{2}\right), \quad n = 0, 1, 2, 3, \dots$$ (c) The wave function for the ground state ($n=0$) is: $$\psi_0(x) = \left(\frac{m\omega}{\pi\hbar}\right)^{1/4} e^{-\frac{m\omega x^2}{2\hbar}}$$ ### Problem 3 (a) To find the normalization constant $A$, we require $\langle \chi | \chi \rangle = 1$: $$|\chi\rangle = |A|^2 \begin{pmatrix} 1-i & 1 \end{pmatrix} \begin{pmatrix} 1+i \\ 1 \end{pmatrix} = |A|^2 ((1-i)(1+i) + 1) = |A|^2 (2+1) = 3|A|^2$$ So, $3|A|^2 = 1$, which gives $A = \frac{1}{\sqrt{3}}$. (b) The probabilities of measuring $S_z = \hbar/2$ and $S_z = -\hbar/2$ are: $$P(S_z = \hbar/2) = \left|\frac{1+i}{\sqrt{3}}\right|^2 = \frac{2}{3}$$ $$P(S_z = -\hbar/2) = \left|\frac{1}{\sqrt{3}}\right|^2 = \frac{1}{3}$$ (c) The expectation value of $S_x$ is: $$\langle S_x \rangle = \langle \chi | S_x | \chi \rangle = \frac{\hbar}{2} \langle \chi | \sigma_x | \chi \rangle$$ where $\sigma_x = \begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix}$. $$\langle S_x \rangle = \frac{\hbar}{2} \frac{1}{3} \begin{pmatrix} 1-i & 1 \end{pmatrix} \begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix} \begin{pmatrix} 1+i \\ 1 \end{pmatrix} = \frac{\hbar}{6} \begin{pmatrix} 1-i & 1 \end{pmatrix} \begin{pmatrix} 1 \\ 1+i \end{pmatrix} = \frac{\hbar}{6} [(1-i) + (1+i)] = \frac{\hbar}{6} (2) = \frac{\hbar}{3}$$ So, $\langle S_x \rangle = \frac{\hbar}{3}$.