Physics · Unit 4 · 5 hrs
Methods of Quantum Mechanics
Exam-focused notes for Methods of Quantum Mechanics (Physics, PHY118): what the TU syllabus asks and how it has actually been tested, with 9 solved past questions from this unit.
What this unit covers
- Schrodinger theory of quantum mechanics and its application
- Outline of the solution of Schrodinger equation for H-atom
- space quantization and spin
- Atomic wave functions
Schrodinger theory of quantum mechanics and its application
Set up Schrodinger equation and discuss the wavefunction. [5]
Consider a free particle moving along the x-direction. The generalized wavefunction is taken as: $$\psi = A e^{i(kx - \omega t)} \quad \cdots (1)$$ where: - $k$ = wave number (propagation constant) - $\omega$ = angular frequency - $A$ = amplitude --- Taking...
Full solved answer →What do you mean by the wavefunction? Discuss its physical significance. Set up time-independent and time-dependent Schrodinger wave equation. What are the implications of this equation? Discuss.[10]
A wavefunction (denoted by Ψ, psi) is a mathematical function associated with a particle moving in a conservative field of force. It describes everything that can be known about a quantum mechanical system. For a particle moving in one dimension, the genera...
Full solved answer →What is the probability of finding a particle in a well of width $a$ at a position $\frac{a}{4}$ from the wall if $n=1$, if $n=2$, if $n=3$. Use the normalized wavefunction $$\psi(x,t) = \left(\frac{2}{a}\right)^{\frac{1}{2}} \sin\left(\frac{n\pi x}{a}\right) e^{-\frac{iEt}{\hbar}}$$ [5]
- Normalized wavefunction: $\psi(x,t) = \left(\dfrac{2}{a}\right)^{1/2} \sin\left(\dfrac{n\pi x}{a}\right) e^{-iEt/\hbar}$ - Well width: $a$ - Position of interest: $x = \dfrac{a}{4}$ - Quantum numbers: $n = 1, 2, 3$ Note on interpretation: The wavefunction...
Full solved answer →What is the probability of finding a particle in a well of width $a$ at a position $\frac{a}{4}$ from the wall if $n=1$, if $n=2$, if $n=3$. Use the normalized wavefunction $$\psi(x) = \left(\frac{2}{a}\right)^{1/2} \sin\left(\frac{n\pi x}{a}\right) e^{-\frac{iEt}{\hbar}}$$ [5]
- Width of well: $a$ - Position: $x = \dfrac{a}{4}$ - Normalized wavefunction: $\psi(x) = \left(\dfrac{2}{a}\right)^{1/2}\sin\left(\dfrac{n\pi x}{a}\right)e^{-iEt/\hbar}$ - Quantum numbers: $n = 1, 2, 3$ Note: The wording asks for "the probability of findin...
Full solved answer →space quantization and spin
A beam of hydrogen atoms is used in a Stern-Gerlach type experiment. The atoms emerge from the oven with a velocity $v = 10^4$ m/sec. They enter a region 20 cm long where there is a magnetic field gradient $\frac{dB}{dz} = 3 \times 10^4$ T/m. The field gradient is perpendicular to the incident velocity of the atoms. The mass of the hydrogen atom is $1.67 \times 10^{-27}$ kg. What is the separation of the two components of the beam as they emerge from the magnet? [5]
Quantity Value ------ Velocity $v = 10^4$ m/s Field region length $L = 0.20$ m Field gradient $dB/dz = 3 \times 10^4$ T/m Mass of H atom $m = 1.67 \times 10^{-27}$ kg Bohr magneton $\muB = 9.274 \times 10^{-24}$ J/T Force on each spin component: $F = \muB \...
Full solved answer →Describe the term 'space quantization'. [5]
Space quantization refers to the quantum mechanical phenomenon in which the orientation of the angular momentum vector of an atom (or any quantum system) in space is not arbitrary but restricted to only certain discrete, allowed directions with respect to a...
Full solved answer →(a) How many atomic states are there in hydrogen with $n=3$? (b) How are they distributed among the sub shells? Label each state with the appropriate set of quantum numbers $n, l, m_l, m_s$. (c) Show that the number of states in a shell, that is, states having the same $n$, is given by $2n^2$. (Hint: $1+2+3+\ldots+n=\frac{n(n+1)}{2}$). [5]
- Principal quantum number: $n = 3$ - Quantum number rules: - $l = 0, 1, \dots, (n-1)$ - $ml = -l, \dots, 0, \dots, +l$ → $(2l+1)$ values - $ms = \pm\tfrac{1}{2}$ → 2 values --- $$\text{Total states} = 2n^2 = 2(3)^2 = 18 \text{ states}$$ --- For $n=3$, $l =...
Full solved answer →(a). How many atomic states are there in hydrogen with n = 3? (b) How are they distributed among the subshells? Label each state with the appropriate set of quantum numbers n, 1, m, m (c) Show that the number of states in a shell, that is, states having the same n, is given by 2n22n^22n2. [5]
- Principal quantum number: $n = 3$ - Quantum number rules: - $l = 0, 1, \ldots, (n-1)$ - $ml = -l, \ldots, 0, \ldots, +l$ → $(2l+1)$ values - $ms = \pm\tfrac{1}{2}$ → 2 values --- For $n = 3$, $l = 0, 1, 2$. $$N = 2n^2 = 2(3)^2 = 18 \text{ states}$$ --- n ...
Full solved answer →Outline of the solution of Schrodinger equation for H-atom
Setup Schrodinger equation for Hydrogen atom using spherical polar coordinates. Separate radial and angular part of this equation using appropriate separation constant. Discuss the separation constant and hence the quantum numbers associated with these two equations. What information can be drawn from the angular part of the Schrodinger equation? Explain.[10]
The hydrogen atom consists of one proton (nucleus) and one electron revolving around it. The potential energy of the electron due to electrostatic attraction is: $$V(r) = -\frac{e^2}{4\pi\epsilon0 r}$$ The time-independent Schrödinger equation in Cartesian ...
Full solved answer →Make Unit 4 stick
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