Quantum Mechanics Notes PDF

# Quantum Mechanics ## What is QM? - It deals with the study of motion of microscopic matter or subatomic particles. ## Why QM? - Classical mechanics couldn't explain the black body radiation. - QM → Quantization of energy (all these theories were explained). ## Why Classical mechanics fails? - C...

# Quantum Mechanics ## What is QM? - It deals with the study of motion of microscopic matter or subatomic particles. ## Why QM? - Classical mechanics couldn't explain the black body radiation. - QM → Quantization of energy (all these theories were explained). ## Why Classical mechanics fails? - Classical mechanics doesn't consider wave character. - Wave function plays a major role and cannot be ignored ## Some experimental evidence or observations - Black Body radiation. - Heat capacity of solids. - Atomic & molecular spectra. - Photoelectric effect. ## Why Classical mechanics failed to experimental results - Classical mechanics doesn't consider wave-particle duality of a particle. - Classical mechanics doesn't consider the wave nature for macroscopic particles. - For macroscopic matter → m is large → A is small → wave nature is ignored. Hence, classical mechanics explanation fits well. - For microscopic matter → m is small → A is large → wave nature is not ignored. Classical mechanics doesn't fit well. - Classical mechanics doesn't take wave character into account ## Wave-particle duality - It is the study of motion of microscopic matter, i.e. particles. - It is the study of motion of subatomic particles. ## Black Body Radiation - A black body is a body which absorbs and emits radiations of all frequencies. - The energy of the black body radiated by a black body at different temperatures per unit volume was calculated and the experimental results for the black body radiation was explained by the Black Body Radiation. ## Wein displacement Law - $A_{max}T=constant$ - It states that the peak of energy density shifts to lower wavelength (blue region) as temperature is increased so the perceived color will be blue. ## Stefar Boltzmann Law - $E\propto T^{4}$ - $E= \sigma T^{4}$ ## Rayleigh-Jeans Law - The first such attempt was made by Rayleigh & Jeans, and they considered electromagnetic field oscillating as electric dipoles. - The presence of radiation as a collection of oscillators of frequency $a$, ## Max Plank's Hypothesis - The radiation emitted by a black body is due to the constituents particles of the black body. - The energy density of a black body at any particular frequency is a multiple of a fundamental quantity called quanta. - $E_{v}dv= 8\pi h\nu^3dv/c^3e^{-\frac{h\nu}{kT}}-1$ - According to Max Planck, Oscillators can only be excited to energies that are integral multiples of $h\nu$. - $E_{v}dv= 8\pi h\nu^3dv/c^3e^{-\frac{h\nu}{kT}}-1$ ## Postulates of Quantum Mechanics - 1. The state of a quantum mechanical system is completely specified by a function $\psi(x, y, z, t)$ that depends upon the coordinates of the system and time. This function is called the wave function or the state function. It has the probability that the particle lies within a volume element $dxdydz$ at time $t$. - 2. The wave function $\psi$ must be single valued, finite, continuous and a well-behaved function. - 3. Corresponding to every variable in classical mechanics there is an operator in quantum mechanics. - 4. An observation of a measurement of the variable associated with an operator A, the only allowed or permissible values are the eigen values 'a' defined by the eigen value equation: $A\psi =a \psi$. - 5. The average or expectation value of the variable A corresponding to the operator A is given by: $<A>=\int\psi^* A\psi d\tau = dxdydz$. - 6. The wave function $\psi$ evolves in time according to time dependent Schrodinger equation. ## Hamiltonian Operator - $E = T + V$ (total energy) - $E = \frac{p^2}{2m} + V$ (kinetic energy + potential energy) - In operator form, $H = \frac{-h^2}{2m}(\frac{\partial^2}{\partial x^2} + \frac{\partial^2}{\partial y^2} + \frac{\partial^2}{\partial z^2}) + V$ ## Schrodinger Wave Equation - $H\psi = E\psi$ - $\frac{-h^2}{2m}(\frac{\partial^2}{\partial x^2} + \frac{\partial^2}{\partial y^2} + \frac{\partial^2}{\partial z^2})\psi +V\psi = E\psi$ ## Formulation of Schrondinger Wave Equation - Schrondinger wave theory is based on the assumption that the behavior of an electron in the atom can be described by an equation similar to the one used to describe standing wave systems such as a vibrating spring fixed at both ends. - The equation of a simple wave in one direction with amplitude $(\psi)$ and velocity $(v)$ is given by: $\psi = A sin(kx-\omega t)$ - In operator form, $H = \frac{-h^2}{2m}(\frac{\partial^2}{\partial x^2} + \frac{\partial^2}{\partial y^2} + \frac{\partial^2}{\partial z^2}) + V$ - $\psi = \psi(x)e^{-2\pi i\nu t}$ - $\frac{\partial^2}{\partial x^2} (\psi(x)e^{-2\pi i\nu t}) = -4\pi^2\nu^2\psi(x) e^{-2\pi i\nu t}$ - The amplitude $\psi$ is a function depending upon variables $x$ and $t$, it can be written as a product of two functions, one dependent on $x$ and the other dependent on $t$. - The total energy $(E) = T + V$ - The total energy $(E) = \frac{1}{2}mu^2 + V$ - $\frac{1}{2}mu^2 = \frac{1}{2m}(E-V)$ - Substituting equation 8 in equation: $\frac{\partial^2}{\partial x^2}(\psi(x)e^{-2\pi i\nu t}) = -4\pi^2\nu^2\psi(x) e^{-2\pi i\nu t}$ - Extending the equation to describe motion in three dimensions, $\psi(x)$ will be replaced by $\psi(x,y,z)$ and $\frac{\partial^2}{\partial x^2}$ will be replaced by $\frac{\partial^2}{\partial x^2}+\frac{\partial^2}{\partial y^2}+\frac{\partial^2}{\partial z^2}$. - $\frac{-h^2}{8\pi^2m}(\frac{\partial^2}{\partial x^2}+\frac{\partial^2}{\partial y^2}+\frac{\partial^2}{\partial z^2})\psi(x,y,z)=-(E-V)\psi(x,y,z)$ - Using de Broglie's equation: $\lambda = \frac{h}{mv}$ - $\frac{-h^2}{8\pi^2m}(\frac{\partial^2}{\partial x^2}+\frac{\partial^2}{\partial y^2}+\frac{\partial^2}{\partial z^2})\psi(x,y,z)=-(E-V)\psi(x,y,z)$ - To solve for the wave function, we need to identify the potential energy function $V$ and solve the differential equation. ## Uncertainty Principle and Wave Particle Duality - **Uncertainty Principle:** This principle states that it is impossible to know both the position and momentum of a particle with absolute certainty. - **Wave Particle Duality:** This is a fundamental concept in quantum mechanics stating that matter has both wave-like and particle-like properties. ## Hermitian Operator - An operator A is said to be Hermitian if $\int\psi_i^* (A\psi_j) d \tau = \int (A \psi_i)^*\psi_i d \tau$ where ${\psi_i}$ and ${\psi_j}$ are the eigenfunctions of A corresponding to two different quantum mechanical states. ## Properties of Hermitian Operator - 1. The eigen values of a Hermitian operator are always real. - 2. Two eigen functions of a Hermitian operator corresponding to different eigen values are orthogonal. - 3. The average or expectation value of a variable associated with a Hermitian operator is always real. ## Angular Momentum Operator - $L = r \times p$ - $L = xp_z - zp_x, yp_x - xp_y, zp_x-xp_z$ ## Commutation of operators for Angular Momentum Operator - 1. $[L_x, L_y] = i \hbar Lz$ - 2. $[L_y, L_z] = i \hbar L_x$ - 3. $[L_z, L_x] = i \hbar L_y$ - 4. $[L^2, L_x] = [L^2, L_y]= [L^2, Lz] = 0$ ## Shift Operator - The operators $L_+$ and $L_-$ are known as step up and step down operators. - $[L^2, L_+]=[L^2, L_-]=0$ ## Particle in one dimensional box - $V= \infty$ at $x=0$ and $x=a$ - $V=0$ at $x=0$ and $x=a$ - $\frac{-h^2}{2m}\frac{\partial^2}{\partial x^2}\psi + V\psi = E\psi$ - $\frac{-h^2}{2m}\frac{\partial^2}{\partial x^2}\psi = E\psi$ - $\frac{\partial^2}{\partial x^2}\psi + \frac{2mE}{h^2}\psi = 0$ - Substituting: $\frac{2mE}{h^2} = k^2$ - $\frac{\partial^2}{\partial x^2}\psi + k^2\psi = 0$ - $\psi = A sinkx+B cos kx$ - $\psi = 0$ at $x=0$ and $x=a$ - $0 = A sin(0) + B cos(0)$ - $B = 0$ - $\psi = A sinkx$ - $\psi = 0$ at $x=a$ - $0 = A sin(ka)$ - $ka = n\pi$ - $k= \frac{n\pi}{a}$ - $E=\frac{n^2h^2}{8ma^2}$ - If $n=0$ then $E=0$ which is not acceptable. - For the above equation, the energy of a particle is quantized. ## Normalization of $\psi$ to determine constant A - For the total probability of finding the particle within a box $= 1$. - $\int\psi^*\psi dx= 1$. ## Orthonormality in 1-D box - $\int\psi_m^*\psi_n dx = \int \frac{2}{a}\sin(\frac{m\pi}{a}x) \frac{2}{a} \sin(\frac{n\pi}{a}x)dx$ $=0$ ## Translational Energy - $\Delta E= (2n+1)\frac{h^2}{8ma^2}$ ## Bohr’s Corresponding Principle - According to this principle, the predictions of quantum mechanics and classical mechanics for a particular system become the same at very high energy values. ## Particle in a 2D box - $V(x,y)=0$ inside the box. - $V(x,y)=\infty$ at the walls and outside the box. - $\frac{-h^2}{2m} (\frac{\partial^2}{\partial x^2} + \frac{\partial^2}{\partial y^2})\psi = E\psi$ - $E = E_x + E_y$ - The wave function $\psi(x,y) = \psi(x)\psi(y)$ - $E = \frac{n_x^2h^2}{8ma^2} + \frac{n_y^2h^2}{8ma^2} $ - $E = \frac{(n_x^2+n_y^2)h^2}{8ma^2} $ ## Degeneracy of energy states - Groups of different quantum states each specified by one set of quantum numbers can have the same energy. Such energy levels, and the corresponding independent states, are said to be degenerate. ## Particle in a 3D box - $V(x, y, z) = 0$ inside the box. - $V(x,y,z)= \infty$ at the wall and outside the box. - $\frac{-h^2}{2m}(\frac{\partial^2}{\partial x^2}+\frac{\partial^2}{\partial y^2}+\frac{\partial^2}{\partial z^2})\psi + V\psi = E\psi$ - $\frac{-h^2}{2m}(\frac{\partial^2}{\partial x^2}+\frac{\partial^2}{\partial y^2}+\frac{\partial^2}{\partial z^2})\psi = E\psi$ - $E = E_x + E_y + E_z$ - $E = \frac{(n_x^2+n_y^2+n_z^2)h^2}{8ma^2}$ ## Simple Harmonic Oscillator - $F = -kx$ - $V=\frac{1}{2}kx^2$ ## Schrodinger equation for Simple Harmonic Oscillator - $\frac{-h^2}{2m}\frac{\partial^2}{\partial x^2}\psi + \frac{1}{2}kx^2\psi =E\psi$ - $\frac{-h^2}{2m}\frac{\partial^2}{\partial x^2}\psi +\frac{1}{2}mk^2x^2\psi =E\psi$ - Defining $a=\frac{2mE}{h^2}$ and $B = (\frac{mk}{h^2})^\frac{1}{4}$ - $\frac{\partial^2}{\partial x^2}\psi + (a-\frac{1}{4}B^4x^2)\psi = 0$ - Substituting $x=B\xi$ - $\frac{\partial^2}{\partial \xi^2}\psi + (a-\frac{1}{4}B^4x^2)\psi = 0$ - $\frac{\partial^2}{\partial \xi^2}\psi + (\alpha-\frac{1}{4}\xi^2)\psi = 0$ - Defining new variable $y = B\xi$ - $\frac{\partial^2}{\partial y^2}\psi + (\alpha-\frac{1}{4}\xi^2)\psi = 0$ - $\frac{\partial^2}{\partial y^2}\psi -(\alpha-\frac{1}{4}y^2)\psi = 0$ ## Hermite Polynomial - The Hermite polynomial of degree $n$ is defined as: $H_n(y)=(-1)^n e^{y^2}\frac{d^n}{dy^n}(e^{-y^2})$ - $H_0(y) = 1$ - $H_1(y) = 2y$ - $H_2(y) = 4y^2-2$ - $H_3(y) = 8y^3 - 12y$ ## Normalized Wave Function for Simple Harmonic Oscillator - $\psi(y) = C_nH_n(y)e^{-\frac{y^2}{2}}$ - $\psi(y) = C_nH_n(\frac{x}{B})e^{-\frac{x^2}{2B^2}}$ ## Energy of Simple Harmonic Oscillator - $E = (n+\frac{1}{2})h\nu$ - $E = (n+\frac{1}{2})\frac{h}{2\pi}\sqrt{\frac{k}{m}} $ ## Zero Point Energy - The lowest possible energy of a simple harmonic oscillator is not zero, but rather a non-zero minimum value. ## Physical Significance of Zero Point Energy - In the ground vibrational state, the molecule vibrates with a minimum energy due to the Heisenberg uncertainty principle. ## Explanation for Bohr’s Correspondence Principle in Light of S.H.O. - According to Bohr’s correspondence principle, classical mechanics and quantum mechanics give the same result for very high energy states. ## Rigid Rotator - A diatomic molecule rotating about an axis perpendicular to the internuclear axis and passing through the center of gravity of the molecules is an example of a rigid rotator. ## Schrodinger Equation for Rigid Rotator - $\frac{-h^2}{2I}(\frac{\partial^2}{\partial \theta^2} + \frac{1}{\sin^2\theta}\frac{\partial^2}{\partial \phi^2})\psi = E\psi$ ## Rotational Quantum Number - $J = 0, 1, 2, 3, \dots$ ## Energy Eigen Values for Rigid Rotator - $E = \frac{h^2}{8\pi^2I}J(J+1)$ - $E = hB J(J+1)$ ## Physical significance of m - $m$ represents the quantization Of $L_z$. ## Writing Hamiltonian Hamiltonian Operator and Schrodinger Eq for a chemical system. - For a one particle system: $H\psi = E\psi$ - $H = \frac{-h^2}{2m}\nabla^2 +V$ - For two particles system, one nucleus and one electron $H\psi = E\psi$ - $H = \frac{-h^2}{2M}\nabla_N^2 + \frac{-h^2}{2m}\nabla_e^2 + V_{ne}$ - For hydrogen atom, two particles: $H\psi = E\psi$ - $H = \frac{-h^2}{2M}\nabla_N^2 + \frac{-h^2}{2m}\nabla_e^2 -\frac{Ze^2}{4\pi\epsilon_0 r}$ ## Quantization of Electronic Energy ## Hydrogen Atom - $V = \frac{Ze^2}{4\pi\epsilon_0 r}$ - $E = E_e + E_t$ (Electronic energy + Translational energy) - $\frac{-h^2}{8\pi^2m}(\nabla^2)\psi+V\psi=E\psi$ ## Center of Mass Coordinates - $X_{cm} = \frac{m_ex_e +m_nx_n}{m_e+m_n}$ - $Y_{cm} = \frac{m_ey_e +m_ny_n}{m_e+m_n}$ - $Z_{cm} = \frac{m_ez_e +m_nz_n}{m_e+m_n}$ ## Coordinates of $e^-$ relative to the nucleus - $x = x_e - x_n$ - $y = y_e - y_n$ - $z = z_e - z_n$ ## Solutions - 1. Centre of Mass Coordinates - 2. Electronic Coordinates. # Molecular Orbital Method - The molecule is considered as a collection of nuclei and then the electrons are assigned to them. # Valency bond Method - The molecule is considered as a collection of atoms, then the interaction between different atoms is considered. ## LCAO-MO Treatment of H₂ molecule ion - The trial wave function in LCAO-MO wave function model is: $\psi_{MO} = c_1\psi_{1sA} + c_2\psi_{1sB}$ ## $H_2^+$ Molecule - $H = \frac{-h^2}{2m}\nabla^2 - \frac{e^2}{4\pi \epsilon_0 r_A}-\frac{e^2}{4\pi \epsilon_0 r_B}+\frac{e^2}{4\pi \epsilon_0 R}$ - $E = <c_1\psi_{1sA}+c_2\psi_{1sB}|H|c_1\psi_{1sA}+c_2\psi_{1sB}>$ - $E= c_1^2<\psi_{1sA}|H|\psi_{1sA}> + c_2^2<\psi_{1sB}|H|\psi_{1sB}> + 2c_1c_2<\psi_{1sA}|H|\psi_{1sB}>$ ## Assumptions: - 1. The Hamiltonian operator belongs to a class of Hermitian operators for which: - $<\psi_{1sA}|H|\psi_{1sB}> = <\psi_{1sB}|H|\psi_{1sA}>$ - 2. Atomic orbtials are normalized: - $\int\psi_{1sA}\psi_{1sA}d\tau = \int\psi_{1sB}\psi_{1sB}d\tau = 1$ ## Coulombs integral - The presence of isolated atoms. - $<\psi_{1sA}|H|\psi_{1sA}> = X$, $<\psi_{1sB}|H|\psi_{1sB}> = B$ ## Resonance integral - It is a measure of the exchange of an electron from one atom to another atom and always has a negative value. - $<\psi_{1sA}|H|\psi_{1sB}> = B$ ## Overlap Integral - It is a measure of the extent of overlapping of the two atomic orbitals. - $<\psi_{1sA}|\psi_{1sB}> = S$ ## Optimizing c, and c₂ by Variation Method - $E(c_1^2+c_2^2+2c_1c_2S) = c_1^2\alpha + c_2^2\beta + 2c_1c_2B$ - $\frac{\partial E}{\partial c_1} = 0$ - $\frac{\partial E}{\partial c_2} = 0$ - $c_1(\alpha-E)+c_2(B-ES) = 0$ - $c_1(B-ES)+c_2(\beta-E) = 0$ - $c_1$ and $c_2$ can't be zero, hence the determinant: - $(\alpha-E)(B-E)-(B-ES)^2 = 0$ - $(\alpha-E)((\beta-E) -(BAB-ESAB)^2 = 0$ - $E_\pm = \frac{\alpha+\beta}{2} \pm \frac{1}{2}\sqrt{(\alpha-\beta)^2+4\epsilon_0^2S^2B}$ - $E_\pm = \frac{\alpha+\beta}{2} \pm \frac{1}{2}\sqrt{(\alpha - \beta)^2+4\epsilon_0^2S^2B}$ - $E_+ = \alpha + B$ - $E_- = \alpha - B$ ## Quantization of Rotational Energy - $E = \frac{h^2}{8\pi^2 I}J(J+1)$ - $E = hBJ(J+1)$ ## Radial Wave Function For H atom