Basic Principles of Quantum Mechanics MCQ Quiz in বাংলা - Objective Question with Answer for Basic Principles of Quantum Mechanics - বিনামূল্যে ডাউনলোড করুন [PDF]

The correct answer is For n = 1: P = 0.031, For n = 2: P = 0.029

EXPLANATION:

P ≈ (2Δx / L) · sin2(nπx / L).

• The formula for the probability is given as:
• Substitute the values for n = 1 and n = 2, x = 0.66L, and Δx = 0.02L:

  P = (2 × 0.02L / L) · sin2(π × 0.66)

  = 0.04 · sin2(0.66π)

  ≈ 0.031.

  P = (2 × 0.02L / L) · sin2(2π × 0.66)

  = 0.04 · sin2(1.32π)

  ≈ 0.029.

  • For n = 1:
    
  • For n = 2:
    

CONCLUSION:

The approximate probabilities are For n = 1: P = 0.031 & For n = 2: P = 0.029

CONCEPT:

Normalization of the Radial Part of a Hydrogen Atom Wave Function

• Normalization ensures that the probability of finding the electron within the entire space is equal to 1.
• The radial wave function for a hydrogen-like atom in its ground state can be written as , where (N) is the normalization constant, and a₀ is the Bohr radius.
• To normalize the wave function, the integral of the probability density over all space must be equal to 1:

CALCULATION:

• The radial wave function is given by .
• To normalize it, we calculate:
• Solving this integral:
  • Using the given solution,
  • This gives us
  • Solving for ( N ), we get:

CONCLUSION:

• The correct normalization constant is:
  • Option (1):

Concept:

In quantum mechanics, the expectation value of energy for an electron in a mixed state can be calculated by taking the weighted average of the energies of each state in the superposition. For the hydrogen atom, the energy of a state is given by:

Energy Formula: The energy of an electron in a hydrogen atom with principal quantum number ( n ) is:

.

In a mixed state, where the wave function is a combination of different states with weights, the expectation value of the energy is calculated as:

, where is the probability of each state and is the energy of each state.

Explanation:

• Given mixed state:

  • ​

• The probability of each state is given by the square of the coefficient:

  • For :

  • For :

  • For :

• Calculate the energy of each state:

  • For ( n = 1 ):

  • For ( n = 2 ):

  • For ( n = 3 ):

• Calculate the expectation value of energy:

  • ​

• Perform the calculations:

• ​Expectation Energy ⟨E⟩ ≈ -3.7 eV

Conclusion:

 The expectation value of energy of this electron in eV will approximately is -3.7

Concept:

The normalization condition requires that the total probability is equal to 1, so the wavefunction must satisfy the following condition:

Explanation:

• Let the wavefunction be written as:

  • ​

• Now, applying the normalization condition:

  • ​

• This gives:

  • ​

• Expanding the square:

  • ​

• Assuming that the wavefunctions are orthonormal:

  • ​

• Thus, the normalization condition becomes:

  • ​

• Solving for c" id="MathJax-Element-54-Frame" role="presentation" style="position: relative;" tabindex="0">cc $c$ :

  • ​

  • ​

• Therefore, the normalized wavefunction is:

  • ​

Conclusion:

The normalized wavefunction is  , which corresponds to option 4.

Concept:

In quantum mechanics, the total angular momentum (L) of a particle in a central potential is an important observable. The eigenvalue of the (L²) operator gives the total angular momentum of the system in terms of the quantum number l.

For a wavefunction (), where the radial part of the wavefunction depends only on r, the angular part corresponds to spherical harmonics (). The eigenvalue of the (L²) operator is:

• Wavefunction in Spherical Coordinates: From the handwritten notes, the wavefunction is written as:

  •  in spherical coordinates.

    • This implies that l = 1 because the wavefunction has a dependence on .

• Total Angular Momentum: Since l = 1, the total angular momentum squared is given by:

  • ​

Explanation: 

• According to the given wavefunction ( ), the wavefunction depends on (), which corresponds to ( l = 1 ). For this value of ( l ), the total angular momentum squared is:

  • ​

Conclusion:

The correct answer is Option 1: ( ), based on the value of ( l = 1 ) for the given wavefunction.

Concept:

The virial theorem relates the average kinetic energy and potential energy in a stable system bound by inverse-square forces, such as the hydrogen atom. The main points of the virial theorem are:​

• Application to Hydrogen Atom: In a hydrogen atom, the electron is bound to the nucleus by an inverse-square Coulomb potential, and the virial theorem applies.

• Kinetic Energy and Potential Energy Relation: According to the virial theorem, the mean kinetic energy of the electron ⟨T⟩ is half of the magnitude of the potential energy ⟨V⟩, with a negative sign.

• Implication for Total Energy: The total energy ⟨E⟩ of the system is related to the potential energy as:

Explanation: 

• According to the virial theorem, the mean kinetic energy ⟨T⟩ of an electron in a hydrogen atom is given by

  • ​

• This relation shows that the kinetic energy is half of the magnitude of the potential energy, but with a negative sign, consistent with the virial theorem for systems bound by inverse-square forces.

Conclusion:

The correct answer is Option 1, which follows from the virial theorem.

The correct answer is 

Explanation:-

Unequal sides
first Excited State
The first excited state will be the next combination, and it must be a combination of quantum numbers that is higher than the first excited state. The possible combinations that we need to check are:
(2,1,1)
(1,2,1)
(1,1,2)

So degeneracy g = 3

∈n.x ny n2 = 

Conclusion:-

So, The energy and degeneracy g, of the second excited state of a particle of mass m moving freely inside a rectangular parallelopiped of sides L, 2L and 3L, are respectively 

The correct answer is 

Explanation:-

ψ = ψ0(x) + ψ1(x)

ψ0(x) → Ground state

ψ1(x) → Exited state

ε0 = 3/2 ℏω , m = 0, 1, 2, 3

ε1 = (2 × 1 + 3/2)ℏω

= 7/2 ℏω

ψ normelized = 

Σ|Ci|2 = 1

= ε |Ci|2 ∈1

= |Ci|2 ∈1 + |C2|2 ∈2

= 

Conclusion:-

So, The energy of the particle is 

The correct answer is 3/2 a0

Concept:-

• Bohr Model & Bohr Radius: While the Bohr model is a simplified view of the atom introduced prior to quantum mechanics, the Bohr radius remains a fundamental constant in describing atomic scales.
• Wavefunction ((ψ)): A mathematical function that describes the quantum state of a particle, including its position in space.
• Probability Density: The square of the wavefunction's magnitude ((|ψ|²)) gives the probability density, which describes how likely it is to find an electron at any given location around the nucleus.

Explanation:- 

The general formula is _{n,l} = \frac{a_0}{2}(3n^2-l(l+1))\)

Putting value of n,l for 1s in above equation

_{1,0} = \frac{a_0}{2}(3(1)^2-0(0+1))\)

_{1,0} = \frac{3a_0}{2}\)

Conclusion:-

So, The average radius for 1s orbital is  3/2 a0

The correct answer is 1.

Concept:- 

• For a particle in a 3D box, the wave function and energy can be represented as below:

and the corresponding energy is:

• The condition for a 3D cubic box is 
• Energy for a 3D cubic box will be:

where: E is the energy, h is Planck's constant, nx, ny, and nz are the quantum numbers associated with the particle (they can be any positive integer), m is the mass of the particle, and L is the length of the box.

CALCULATION:

• Given that the energy is four times the lowest energy level:
  • Lowest energy, .
  • Thus, the target energy is .
• To achieve this energy level, the quantum numbers must satisfy:
  • .
  • Possible combinations are:
    • (2, 1, 1)
    • (1, 2, 1)
    • (1, 1, 2)

The correct answer is: Option 4 - Degeneracy is 3