All exercise S.Qs of identical particles of Quantum Mechanics. We have gathered and arranged all S.Qs for BS/MSc Physics students.
Q.1.1 Differentiate between bosons and fermions.
Answer: Bosons:
- They have integral spin. $S=0,1\hbar ,2\hbar ,3\hbar…$
- They do not obey the Pauli Exclusion Principle.
- They obey Bose – Einstein statistics.
- They are described by symmetric wave functions ($\mathrm{\Psi}=+\mathrm{\Psi})$
- They create interaction between matter(e.g. energy)
- Examples: Higgs Bosons, photons, gluon , graviton, W+,W– ,Z etc.
Fermions:
- They have half – odd integral spin. $S=0,1\hbar /2,3\hbar/2 ,5\hbar/2…$
- They obey the Pauli Exclusion principle.
- They obey Fermi-Dirac statistics.
- They are described by anti-symmetric wave functions(Ψ = -Ψ)
- They make all the matter in the universe.
- Examples: electrons, proton, neutron, quarks, neutrino etc.
Q.1.2 How to differentiate the identical particle?
Answer: Particles with same mass, spin, charge and magnetic moment are called identical particles. These particles that cannot be distinguished from one another, even in principle, There are two main categories of identical particle “Bosons and Fermions”
Q.1.3 why we cannot apply the exchange operator at one particle?
Answer: Exchange operator interchange the position of (1- particle to 2-particle) ith and jth particles, so one particle is not need to change the position and we cannot apply exchange operator at one particle.
Q.1.4 what is the effect of coulomb potential under permutation.
Answer: The coulomb potential which results from electron-electron and electron-nucleus interaction. It is invariant under the exchange of any pair of electron. The Hamiltonian also invariant (symmetric) under permutations.
$$V\ \left(\vec{r_{1\ }},\vec{r_2},\ \vec{r_3},\ldots\ \vec{r_z}\right)=-\sum_{i=1}^{z}{\frac{{Ze}^2}{\left|\vec{r_i}-\vec{R}\right|}+\ \sum_{i>j}\frac{e^2}{\left|\vec{r_i}-\vec{r_j}\right|}}$$
Q.1.5 Can more than two fermions occupy the same state?
Answer: No more than two fermions occupy the same state, according to Pauli Exclusion Principle that two electrons cannot occupy simultaneously the same quantum state having same spin.
Q.1.6 why bosons cannot satisfy the Pauli Exclusion Principle?
Answer: Bosons particles with an integer spin are not subject to the Pauli Exclusion Principle any number of identical bosons can occupy the quantum state. The Pauli Exclusion Principle states that no two electrons can have the same quantum state.
Q.1.7 Justify 4He is boson and 3He is fermion?
Answer: 4He: It is an alpha particle, consists of four nucleons (two protons and two neutrons). It has an even number of spin. 4He = ($\frac{\hbar}{2}$ + $\frac{\hbar}{2}$ + $\frac{\hbar}{2}$) =$\frac{4\hbar}{2}$= $2\hbar$ . So it is called bosons.
3He: It consists of three nucleons (one neutron and two protons). It has an odd number of spin. 3He = ($\frac{\hbar}{2}$ + $\frac{\hbar}{2}$ + $\frac{\hbar}{2}$)= $\frac{3\hbar}{2}$. So it is called fermions.
Q.1.8 why atoms behave like waves at absolute temperature?
Answer: (due to decrease in energy) and atoms come closer. They further suggested that, if the temperature furthers slow down up to zero kelvin (0 k), which is lowest possible temperature. These slow vibrating atoms are accumulating and transform themselves into wave like behavior.
Q.1.9 Differentiate between 1st and 2nd quantization.
Answer: 1st quantization: General form of quantum mechanics is denoted by first quantization. In case we have a fixed and finite number of degree of freedom.
2nd quantization: we discuss the interaction between many particles by introducing ladder operators. Thus, it is such a method in which particles may be created or destroyed. Second quantization formulism is based on knowledge of possible quantum states and their occupation by particles.
Q.1.10 what is FOCK space?
Answer: The sum of space with no particle |0>, , the space with one particle and so on. Thus sum of space is known as FOCK space.
FOCK space $= |0>+|1>+|2>+\ldots$
Q.1.11 Define following terms with examples: Bosons, Fermions, Constant of motion
Answer: Bosons: Particles with integral spin such as pions, alpha particles, photons etc. are called bosons.
Fermions: Particles with half odd integral spin such as electrons, quarks, positrons, neutrons, protons etc. are called fermions.
Constant of motion: The operator corresponding to a physical quantity is said to be constant of motion if it commutes with Hamiltonian. For example, parity operator is constant of motion because,
$\left[{\hat{P}}_{ij},\ \hat{H}\right]=0$
Q.1.12 what is Bosons condensation?
Answer: Pauli Exclusion Principle is only applicable to fermions. There is no restriction on number of bosons that can occupy a single state. Bosons tend to condense in same state, the ground state; this is called boson condensation. For example all particles of a boson system liquid ${^4}He$ occupy same ground state.
Q.1.13 Determine the ground state of carbon atom 6C.
Answer: Ground state configuration of carbon atom 6C is,
$$1s^22s^22p^2$$
Hence its total angular momentum is determined by 2p electrons. The coupling of two spins s=1/2 gives two values for their total spin S=0 or S=1 Coupling of two individual orbital angular momenta l=1 gives three values for total angular momentum L = 0, 1 or 2. But Pauli Principle tells us that total wave function has to be anti-symmetric that is spin and orbital parts of wave function must have opposite symmetries, since singlet spin state S = 0 is anti-symmetric, the spin triple S = 1 is symmetric, the orbital triple L = 1 is anti-symmetric, the orbital state L = 2 is symmetric, orbital singlet state L = 0 is symmetric, the following states are anti-symmetric:
$${^1}S_0,\ \ \ \ \ \ \ {^3}P_0,\ \ \ \ \ {^3}P_1,\ \ \ {^3}P_2,\ \ \ or\ \ \ \ {^1}D_2$$
Hence any one of these states can be ground state of carbon.
Q.1.14 what is symmetrization postulate?
Answer: According to this postulate,
“The states of systems containing N identical particles are either totally symmetric or totally anti-symmetric under interchange of any pair of particles and that states with mixed symmetry do not exist”
Besides that, this principle has following two important properties:
- Particles with integral spin are called bosons and have symmetric states.
- Particles with half odd integral spin are called fermions and have anti-symmetric states.
Q.1.15 Consider a system of one kind of non-interacting particles. Let their momentum be same and their spin is equal to 1/2. If these particles did not possess spin we could describe the system by “pure case” ensemble. However, we do not know whether the spins of all the particles are parallel. Is it possible to use experiment of Stern Gerlach type to determine whether this beam of particles correspond to a pure case or to a mixed ensemble?
Answer: Yes, in case of mixed ensemble for every direction of inhomogeneous magnetic field we shall always get a splitting into two beams. In case of pure ensemble by a suitable alignment of the instrument we can obtain the disappearance of one of beams.
Q.1.15 Given qualitative reasons that why spin angular momentum commute with position and linear momentum operators.
Answer: Since spin does not depend upon spatial degrees of freedom, the components of spin angular momentum commute with all spatial operators, notably the orbital angular momentum, position and linear momentum operators i.e.
$$\left[{\hat{S}}_j,\ {\hat{L}}_k\right]=0.\ \ \ \ \ \ \left[{\hat{S}}_j,\ {\hat{r}}_k\right]=0,\ \ \ \ \ \ \left[{\hat{S}}_j,\ {\hat{p}}_k\right]=0$$
