問題一覧
1
The most probable radial distance of the electron in the ground state of hydrogen is:
B.r=a0
2
Heisenberg's Uncertainty Principle states that the uncertainties in position (Ax) and momentum (Ap) of a particle satisfy the inequality: Axßp 2 C/2. What is the correct value of C?
C. h/2
3
If a wavefunction (x) is an eigenfunction of an operator A with eigenvalue a, what is the expectation value <A>?
A. a
4
he relative Schrödinger equation resembles the equation for a single particle in a central force field. Which physical problem does this most closely correspond to?
B. The hydrogen atom problem.
5
For atoms with more than three electrons, which of the following effects contributes the most to the breakdown of the simple hydrogen-like energy level structure?
B) Electron-electron interactions and screening effects
6
What is the primary reason the zero-order approximation overestimates the binding energy of the helium atom?
C) It assumes electrons do not interact with each other
7
Why must the total wavefunction of a multi-electron system be antisymmetric with respect to the exchange of two electrons?
C) Because electrons are fermions and obey the Pauli Exclusion Principle
8
Which of the following selection rules applies to the angular momentum quantum numbers in electric dipole transitions for helium?
A) deta L = 0 , +- (but not L = 0 -> L = 0 )
9
The number of radial nodes in the hydrogen atom wavefunction is given by:
C. n-1-1
10
How does the zero-order approximation predict the spatial probability distribution of the two electrons in the helium atom?
C. The electrons independently follow a 1s-orbital distribution similar to hydrogen
11
In the zero-order approximation, which of the following best describes the first excited state of the helium atom?
A) One electron remains in the 1s orbital while the second electron is excited to a higher orbital
12
Which of the following best explains the physical significance of Heisenberg's Uncertainty Principle?
A. A particle's position and momentum cannot be measured simultaneously with arbitrary precision because measurement disturbs the system.
13
For a hellum-like atom with nuclear charge Z, what is the approximate energy of a single electron in the zero-order approximation?
B)-Z/n^2 RH
14
In the zero-order approximation, which of the following best describes the total orbital angular momentum L of the helium atom in its ground state?
A) L = 0
15
Which of the following expressions represents the total zero-order energy of the helium atom assuming a hydrogen-like model?
C) E=-8RH
16
Which of the following best explains why the Aufbau principle correctly predicts electron configurations for most multielectron atoms?
A) It assumes that each electron moves in an average field created by all other electrons.
17
Which quantum numbers fully describe each electron in the helium atom under the zero-order approximation?
C) The principal (n), azimuthal (1), magnetic (ml), and spin (ms) quantum numbers
18
Which of the following is true about the parahelium and orthohelium excited states in the zero-order approximation?
C. Parahelium has electrons in a singlet spin state, while orthohelium has electrons in a triplet spin state
19
Which of the following statements is true regarding the degeneracy of hydrogen atom energy levels?
A. Energy levels depend only on n, and states with different I values but the samen are degenerate.
20
How does electron-electron repulsion modify the excited-state energy levels in helium compared to the zero-order approximation?
C) It splits degenerate states into different energy levels
21
In the zero-order approximation, the wavefunction of a helium-like atom is constructed as:
A. A simple hydrogenic wavefunction for each electron, neglecting electron- electron interactions
22
The third postulate of quantum mechanics states that the only possible result of measuring an observable is one of the eigenvalues of its corresponding operator. If a wavefunction W is expressed linear combination of eigenfunctions of a Hermitian operator A, what happens when A is measured?
A. The wavefunction collapses into a single eigenfunction corresponding to one of the eigenvalues.
23
The commutator of two quantum mechanical operators, (A,B) AB-BA, provides insight into their simultaneous measurement. What is the physical significance of the commutator being nonzero?.
C. The uncertainty principle applies, meaning that measuring one observable precisely increases uncertainty in the other.
24
In a helium excited state where one electron is in 1s and the other is in 3d, what are the possible total angular momentum quantum numbers J (considering spin-orbit coupling)?
D) J = 2, 3, 4
25
Which of the following statements is incorrect regarding quantum observables and their associated operators?
C. Any function of a Hermitian operator is alto Hermitian.
26
Suppose an observable A is measured on a system, and the result is found to be an. If a second measurement of A is performed immediately after the first, what is the expected outcome?
A. The same eigenvalue a n with probability 1.
27
What is the primary quantum mechanical reason that two electrons in the same orbital must have opposite spins?
B) To satisfy the Pauli Exclusion Principle
28
The Schrödinger equation for a two-particle system with masses m1 and m2 in the absence of external forces can be rewritten using which key Transformation?
A. Separation into center-of-mass and relative motion coordinates.
29
Which of the following correctly represents the Pauli Exclusion Principle for a multi-electron wavefunction?
C) The total wavefunction must be antisymmetric under the exchange of two electrons
30
According to the first postulate of quantum mechanics, which of the following statements correctly describes the nature of a quantum state?
B. The state of a quantum system is represented by a wavefunction, which contains all possible measurements of the system.
31
The quantum numbers n,l,m, arise from solving the Schrödinger equation for the hydrogen atom. What is the physical significance of the quantum number 1?
A. It determines the shape of the orbital.
32
The radial wave function R(r) for the hydrogen atom satisfies the radial Schrödinger equation. Which of the following correctly describes its behavior at large distances r?
C. It decays exponentially.
33
According to Postulate 5 of quantum mechanics, what happens to a system's wavefunction immediately after a measurement of an observable A is made and an eigenvalue a, is obtained?
A. The wavefunction collapses into the eigenfunction on corresponding to an.
34
For an excited state of helium where one electron is in is and the other is in 2p, what are the possible values of the total orbital angular momentum quantum number L?
D) L = 1,2
35
Which of the following statements about expectation values is incorrect?
C. The expectation value of an operator is given by (A)»Ψ-AWdx, even if W is not normalized.
36
The quantum mechanical operator for an observable A is given by A. Which of the following conditions must be satisfied for å to correspond to a physically measurable quantity?
C. A must be a Hermitian operator.
37
If a system is initially in a state W that is a linear combination of eigenfunctions , of an observable Â, what determines the probability of obtaining a specific eigenvalue a, upon measurement?
B. The square of the expansion coefficient (cn) in the eigenfunction expansion of W.
38
In a multielectron atom, which quantum number determines the energy splitting within a given principal quantum number (n) shell in the absence of external fields?
B) Orbital angular momentum quantum number I
39
How does the screening effect influence the energy levels of a helium-like atom when compared to the zero-order approximation?
B) It decreases the effective nuclear charge felt by the electrons
40
A quantum particle is confined in a one-dimensional box of length L. According to Heisenberg's Uncertainty Principle, what can be said about the minimum possible momentum uncertainty (Ap) of the particle?
Α. Δρ >_h/2L
41
In the zero-order approximation, what is the energy of the first excited state of helium, assuming one electron is in the is orbital and the other is in the 2s orbital?
B)-3.25RH
42
In quantum mechanics, which of the following statements regarding mathematical operators and mechanical variables is incorrect?
D. The Hamiltonian operator always commutes with the momentum operator in all systems.
43
In the center-of-mass and relative motion transformation, which of the following statements is true about the center-of-mass motion?
B. It behaves as a free-particle motion.
44
The radial wavefunction R(r) of the hydrogen atom is a solution to the radial part of the Schrödinger equation. What does the radial node of R(r) represent?
B. A spherical surface where the probability density is zero.
45
Which of the following orbitals experiences the strongest shielding effect in atoms with more than three electrons?
C) 3d
46
If a quantum system is initially in a non-eigenstate of an observable Å, what happens to the wavefunction immediately after the measurement of A?
A. The wavefunction collapses into an eigenstate of Å associated with the measured eigenvalue.
47
In the case of two electrons in the same atom, what is the consequence of having an antisymmetric spin wavefunction?
A) The spatial wavefunction must be symmetric
48
If an observable A is measured in a quantum system, what is the mathematical requirement for the wavefunction W to yield a definite value of A with 100% certainty?
A. W must be an eigenfunction of A with a corresponding eigenvalue
49
In the zero-order approximation for helium-like atoms, which quantum number determines the probability distribution of each electron?
A) Principal quantum number (n)
50
Which of the following transitions is forbidden in helium under the selection rules for electric dipole transitions?
C) 1s2s→ 1s2
51
In the zero-order approximation, the two electrons in the helium atom are assumed to move independently in which type of potential?
C) A hydrogen-like Coulomb potential due to the nucleus
52
The second postulate of quantum mechanics states that every observable quantity is associated with a Hermitian operator. What is the significence of this condition?
A. Hermitian operators guarantee that eigenvalues correspond to real physically measurable quantities.
53
Which of the following commutators is directly responsible for Heisenberg's position-momentum uncertainty relation?
B. [x^p^x]=ih
54
In the zero-order approximation, what is the primary reason why atoms with more than three electrons do not strictly follow a hydrogen-like model?
B) Electron-electron repulsions significantly modify energy levels.
55
Which of the following correctly describes the relationship between electron indistinguishability and exchange energy?
B) Exchange energy results from the antisymmetry of the total wavefunction and is a purely quantum mechanical effect
56
Which of the following correctly describes the total spin angular momentum S for the helium atom in an excited state?
A) s = 0 for singlet states and s = 1 for triplet states
57
Which of the following best describes why the zero-order approximation for helium-like atoms does not accurately predict the ground-state energy
C) It ignores electron-electron repulsion