solve this question---Here is the extracted content from the image: **Question 1:** 1. Describe, in your own words, what is Quantum Mechanics (QM), what kind of objects does QM deal with? At which length scales does the quantum mechanical description become important? Why is QM important for Chemistry and Engineering? How does QM differ from Classical Mechanics? (give at least 4 differences) **Question 2:** 2. Consider a particle moving in 1D along $x \in (-\infty, +\infty)$. The particle has no spin. Describe, in your own words, how do you perform the passage from Classical Mechanics into Quantum Mechanics. Write down the corresponding procedure mathematically, and build up in this manner the time dependent Schr\"odinger equation (TDSE). Explain, in your own words, what is the TDSE good for. Write down the corresponding time independent Schr\"odinger equation (TISE), and explain what are the associated eigenvalues of $\hat{H}$ good for. Explain how can solutions of the TISE be used to solve TDSE. **Question 3:** 3. Consider a particle of mass $m$ moving on a circle (radius $R$). The particle has no spin. Write down the corresponding Hamiltonian $\hat{H}$ and the angular momentum operator $\hat{L}$. Solve the eigenvalue problems of $\hat{L}$ and $\hat{H}$ (eigenvalues and unit normalized eigenfunctions), and represent the obtained eigenvalue spectra graphically. Write down the orthonormality relations for the eigenfunctions. Write down the time dependent Schr\"odinger equation (TDSE) for our particle on the ring, and explain how can the just constructed eigenfunctions be used to propagate a given $(t=0)$ initial state in time. **Question 4:** 4. Consider a particle moving along $x \in (-\infty, +\infty)$, without spin. Let the quantum state if this particle be described by wavefunction $\psi(x)$. Explain what it means when we say that $\psi(x)$ must be square integrable. Describe how do we normalize a given square integrable wavefunction $\psi(x)$ (which is given to you without normalization). Suppose now that $\psi(x)$ is unit normalized. What is the meaning of $|\psi(x)|^2$? How do we determine the probability of finding our particle between $x = -7$ and $x = +3$? Explain how do we get from $\psi(x)$ an information about the momentum probability density. Write down the definitions of the mean momentum $\bar{p}$ and the standard deviation $(\Delta p)$. **Question 5:** 5. Consider a particle moving along $x \in (-\infty, +\infty)$, without spin. Write down the kinetic energy operator $\hat{T}$, and consider also the potential energy operator $\hat{V} = \hat{V}(x)$. Determine the commutator $[\hat{T}, \hat{V}]$. Prove that the expectation value of the kinetic energy of our particle is always positive. Prove that the expectation value of the potential energy of our particle is always larger than the minimum $V_0$ of the potential. Prove that the ground state energy $\hat{H} = \hat{T} + \hat{V}$ is always larger than $V_0$. Explain what is meant by the zero point energy (ZPE).

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Quantum Mechanics is a fundamental theory that describes the physical properties of nature at atomic and subatomic scales. It deals with microscopic objects like electrons, protons, neutrons, photons, atoms, and molecules. The quantum mechanical description becomes important at nanometer scales, typically ten to the minus nine meters and smaller. QM is crucial for Chemistry as it explains chemical bonds and reactions, and for Engineering it enables technologies like semiconductors, lasers, and quantum computing. Quantum Mechanics differs from Classical Mechanics in four fundamental ways. First, quantization: in QM energy levels are discrete, while classical mechanics allows continuous values. Second, wave-particle duality: QM shows that particles exhibit wave properties, whereas classical mechanics treats particles and waves as distinct entities. Third, probability: QM predicts probabilistic outcomes rather than the deterministic results of classical mechanics. Fourth, the uncertainty principle: QM has fundamental limits on measurement precision, while classical mechanics assumes perfect precision is theoretically possible. The transition from Classical to Quantum Mechanics follows a systematic quantization procedure. First, we identify the classical Hamiltonian H of x and p, which equals p squared over two m plus V of x. Second, we replace classical observables with quantum operators: position x becomes the operator x hat, and momentum p becomes the operator minus i h-bar partial over partial x. Third, this gives us the quantum Hamiltonian operator. The time evolution is then governed by the Time-Dependent Schrödinger Equation, which states that i h-bar partial psi over partial t equals H hat psi. For Question 3, we consider a particle of mass m moving on a circle of radius R. The Hamiltonian operator is H hat equals L hat squared over two m R squared, where L hat is the angular momentum operator equal to minus i h-bar partial over partial phi. Solving the eigenvalue problems, we find that the angular momentum eigenvalues are L n equals n h-bar for integer n from minus infinity to plus infinity. The energy eigenvalues are E n equals h-bar squared n squared over two m R squared. The normalized eigenfunctions are psi n of phi equals one over square root of two pi times e to the i n phi. The orthonormality relation shows that the integral from 0 to 2 pi of psi m star times psi n d phi equals the Kronecker delta. The energy spectrum shows discrete levels with degeneracy for non-zero n values, since positive and negative n give the same energy. To summarize what we've learned about Quantum Mechanics: First, QM describes microscopic systems with discrete energy levels and probabilistic outcomes, fundamentally different from classical mechanics. Second, the transition from classical to quantum involves replacing observables with operators and describing states with wavefunctions. Third, the Schrödinger equation governs quantum evolution, with eigenvalues representing the allowed energy levels of the system. Fourth, the particle on a ring example demonstrates how angular momentum and energy become quantized with discrete spectra. Finally, QM provides the theoretical foundation for chemistry, engineering, and modern quantum technologies like semiconductors and quantum computing.