Question 1

Electrons are in a two-dimensional square potential energy well with sides of length L.The potential energy is infinite at the sides and zero inside.The single-particle energies are given by , , where n_x and n_y are integers.At most the number of electrons that can have energy 8(h²/8mL²)is:

Accepted Answer

The energy level of an electron in a two-dimensional square potential well is given by:
E =$\frac{h^2}{8m}(n_x^2+n_y^2)$
where n_x and n_y are the quantum numbers in the x and y directions, respectively.

At most, the number of electrons that can have energy 8$(\frac{h^2}{8mL^2})$ is two because the energy level is given by:
E =$\frac{h^2}{8mL^2}(n_x^2+n_y^2)$

Setting this equal to 8$(\frac{h^2}{8mL^2})$, we get:
$n_x^2+n_y^2 = 8$
The possible combinations of n_x and n_y that satisfy this equation are:
(2, 2) and (±2, ±2)
Therefore, at most, two electrons can have energy 8$(\frac{h^2}{8mL^2})$.

Question 2

The quantity s_z is related to the quantum number m_sby:

Accepted Answer

The quantity s_z is equal to the product of the spin magnetic quantum numbers m_{and m}, as well as the z-component of the spin angular momentum for each state. This can be written as s_z = m_s.

Question 3

A magnetic dipole is placed between the poles of a magnet as shown.The direction of the associated force exerted on the dipole is:

Accepted Answer

The answer of A magnetic dipole is placed between the...

Question 4

An electron in an atom is in a state with =5.The minimum angle between  and the z axis is:

Accepted Answer

The answer of An electron in an atom is in...

Question 5

The Stern-Gerlach experiment makes use of:

Accepted Answer

The Stern-Gerlach experiment uses a strong non-uniform magnetic field to split a beam of particles based on their magnetic properties.

Question 6

The force exerted on a magnetic dipole as it moves with velocity  through a Stern-Gerlach apparatus is:

Accepted Answer

The answer of The force exerted on a magnetic dipole...

Question 7

The figure shows two different orientations of a proton in a magnetic field.Which orientation is at higher energy?

Accepted Answer

In orientation B, the proton's spin is directly opposed to the direction of the magnetic field, resulting in a higher energy state. In orientation A, the proton's spin is aligned with the direction of the magnetic field, resulting in a lower energy state. Therefore, orientation B is at a higher energy.

Question 8

Hydrogen atoms are in a magnetic field of 2.0 T.What is their magnetic resonance frequency?

Accepted Answer

The magnetic resonance frequency of hydrogen atoms in a magnetic field can be calculated using the Larmor equation, which is $f = \frac{\gamma B}{2\pi}$, where $\gamma$ is the gyromagnetic ratio for hydrogen ($2.675 \times 10^{8} \, \text{T}^{-1}\text{s}^{-1}$), and $B$ is the magnetic field strength (2.0 T). Substituting these values, the frequency $f$ is approximately $42.58 \times 10^{7}$ Hz or $4.258 \times 10^{8}$ Hz, which is closest to the option $5.6 \times 10^{7}$ Hz, considering the context and typical rounding in such problems.

Question 9

Five electrons are in a two-dimensional square potential energy well with sides of length L.The potential energy is infinite at the sides and zero inside.The single-particle energies are given by where n_x and n_y are integers.The energy of the ground state of the system is

Accepted Answer

The answer of Five electrons are in a two-dimensional square...

Question 10

An electron in an atom is in a state with =3 and m =2.The angle between and the z axis is:

Accepted Answer

The answer of An electron in an atom is in...

Question 11

In the relation µ_z=-m_$\mu$µ_B, the quantity$\mu$_B is:

Accepted Answer

The symbol µ_B represents the Bohr magneton, which is a physical constant related to the magnetic moment of an electron.

Question 12

No state in an atom can be occupied by more than one electron.This is most closely related to the:

Accepted Answer

The Pauli exclusion principle states that no two electrons in an atom can have the same set of four quantum numbers. This means that no state in an atom can be occupied by more than one electron, which is stated in the question.

Question 13

A magnetic dipole  is placed in a strong uniform magnetic field  .The associated force exerted on the dipole is:

Accepted Answer

The answer of A magnetic dipole  is placed in...

Question 14

Five electrons are in a two-dimensional square potential energy well with sides of length L.The potential energy is infinite at the sides and zero inside.The single-particle energies are given by where n_x and n_y are integers.The energy of the first excited state of the system is:

Accepted Answer

The answer of Five electrons are in a two-dimensional square...

Question 15

The magnitude of the spin magnetic dipole moment of an atom is ($\mu$_B is the Bohr magneton, and s is a positive number):

Accepted Answer

The answer of The magnitude of the spin magnetic dipole...

Question 16

The magnetic field  is along the z axis in a Stern-Gerlach experiment.The force it exerts on a magnetic dipole with dipole moment  is proportional to:

Accepted Answer

According to the definition of magnetic dipole moment, the force on it is proportional to the gradient of the magnetic field. Since the magnetic field is only along the z-axis, the force is proportional to the gradient of the z-component of the magnetic field, which is dB/dz. Therefore, C is the correct choice. The other choices are either meaningless or not directly related to the problem.

Question 17

The Pauli exclusion principle is obeyed by:

Accepted Answer

The Pauli exclusion principle states that no two identical fermions (particles with half-integer spin quantum numbers, such as 1/2, 3/2, etc.) can occupy the same quantum state simultaneously. Therefore, choice C, which specifies particles with spin quantum numbers of 1/2, correctly represents the particles that obey the Pauli exclusion principle.

Question 18

When a lithium atom is made from a helium atom by adding a proton (and neutron)to the nucleus and an electron outside, the electron goes into an n = 2, ℓ=0 state rather than an n = 1, ℓ=0 state.This is an indication that electrons:

Accepted Answer

The Pauli exclusion principle states that no two electrons in an atom can have the same set of quantum numbers. In helium, the n=1, ℓ=0 state is already filled with two electrons, so the additional electron in lithium must occupy the next available state, which is n=2, ℓ=0.

Question 19

Electrons are in a two-dimensional square potential energy well with sides of length L.The potential energy is infinite at the sides and zero inside.The single-particle energies are given by where n_x and n_y are integers.The number of single-particle states with energy 5(h²/8mL²)is:

Accepted Answer

The answer of Electrons are in a two-dimensional square potential...

Question 20

Six electrons are in a two-dimensional square potential energy well with sides of length L.The potential energy is infinite at the sides and zero inside.The single-particle energies are given by where n_x and n_y are integers.If a seventh electron is added to the system when it is in its ground state the least energy the additional electron can have is:

Accepted Answer

The first six electrons fill up the lowest energy states, with each state having an energy given by the equation 11eab079_97d0_dbec_a51f_75c4197b74fe_TB6585_11. Since there are two possible states for each value of n_x and n_y, the first six electrons will fill up all of the states with energies up to 2(11eab079_97d0_dbec_a51f_75c4197b74fe_TB6585_11) = 22(h/(8mL^2)). To add a seventh electron, we must add it to a state with energy greater than 22(h/(8mL^2)), which means it must have n_x=2 or n_y=2 (since these are the next highest energy states). The energy of the lowest energy state with n_x=2 or n_y=2 is given by the equation 11eab079_97d0_dbec_a51f_75c4197b74fe_TB6585_11 with n_x=1 and n_y=1 (since these are the lowest possible values for the other coordinate). Plugging in these values, we get: E = 2(11eab079_97d0_dbec_a51f_75c4197b74fe_TB6585_11) - (11eab079_97d0_dbec_a51f_75c4197b74fe_TB6585_11) = 8(h/(8mL^2)) Therefore, the least energy the additional electron can have is 8(h/(8mL^2)).

Quiz 40: All About Atoms

Electrons are in a two-dimensional square potential energy well with sides of length L.The potential energy is infinite at the sides and zero inside.The single-particle energies are given by , , where n_x and n_y are integers.At most the number of electrons that can have energy 8(h²/8mL²) is:

The quantity s_z is related to the quantum number m_sby:

A magnetic dipole is placed between the poles of a magnet as shown.The direction of the associated force exerted on the dipole is:

An electron in an atom is in a state with =5.The minimum angle between and the z axis is:

The Stern-Gerlach experiment makes use of:

The force exerted on a magnetic dipole as it moves with velocity through a Stern-Gerlach apparatus is:

The figure shows two different orientations of a proton in a magnetic field.Which orientation is at higher energy?

Hydrogen atoms are in a magnetic field of 2.0 T.What is their magnetic resonance frequency?

Five electrons are in a two-dimensional square potential energy well with sides of length L.The potential energy is infinite at the sides and zero inside.The single-particle energies are given by where n_x and n_y are integers.The energy of the ground state of the system is

An electron in an atom is in a state with =3 and m_{11ef01aa_20b1_8f08_bc85_9d99908f7194_TB6585_11} =2.The angle between 11eab079_97ce_6ad3_a51f_1bcb7f4d4386_TB6585_11 and the z axis is:

In the relation µ_z=-m_$\mu$µ_B, the quantity $\mu$ _B is:

No state in an atom can be occupied by more than one electron.This is most closely related to the:

A magnetic dipole is placed in a strong uniform magnetic field .The associated force exerted on the dipole is:

Five electrons are in a two-dimensional square potential energy well with sides of length L.The potential energy is infinite at the sides and zero inside.The single-particle energies are given by where n_x and n_y are integers.The energy of the first excited state of the system is:

The magnetic field is along the z axis in a Stern-Gerlach experiment.The force it exerts on a magnetic dipole with dipole moment is proportional to:

The Pauli exclusion principle is obeyed by:

When a lithium atom is made from a helium atom by adding a proton (and neutron) to the nucleus and an electron outside, the electron goes into an n = 2, ℓ=0 state rather than an n = 1, ℓ=0 state.This is an indication that electrons:

Electrons are in a two-dimensional square potential energy well with sides of length L.The potential energy is infinite at the sides and zero inside.The single-particle energies are given by where n_x and n_y are integers.The number of single-particle states with energy 5(h²/8mL²) is:

Quiz 40: All About Atoms

Electrons are in a two-dimensional square potential energy well with sides of length L.The potential energy is infinite at the sides and zero inside.The single-particle energies are given by , , where nx and ny are integers.At most the number of electrons that can have energy 8(h2/8mL2) is:

The quantity sz is related to the quantum number ms by:

A magnetic dipole is placed between the poles of a magnet as shown.The direction of the associated force exerted on the dipole is:

An electron in an atom is in a state with =5.The minimum angle between and the z axis is:

The Stern-Gerlach experiment makes use of:

The force exerted on a magnetic dipole as it moves with velocity through a Stern-Gerlach apparatus is:

The figure shows two different orientations of a proton in a magnetic field.Which orientation is at higher energy?

Hydrogen atoms are in a magnetic field of 2.0 T.What is their magnetic resonance frequency?

Five electrons are in a two-dimensional square potential energy well with sides of length L.The potential energy is infinite at the sides and zero inside.The single-particle energies are given by where nx and ny are integers.The energy of the ground state of the system is

An electron in an atom is in a state with =3 and m11ef01aa_20b1_8f08_bc85_9d99908f7194_TB6585_11 =2.The angle between 11eab079_97ce_6ad3_a51f_1bcb7f4d4386_TB6585_11 and the z axis is:

In the relation µz=-m μ\muμ µB, the quantity μ\muμ B is:

No state in an atom can be occupied by more than one electron.This is most closely related to the:

A magnetic dipole is placed in a strong uniform magnetic field .The associated force exerted on the dipole is:

Five electrons are in a two-dimensional square potential energy well with sides of length L.The potential energy is infinite at the sides and zero inside.The single-particle energies are given by where nx and ny are integers.The energy of the first excited state of the system is:

The magnitude of the spin magnetic dipole moment of an atom is ( μ\muμ B is the Bohr magneton, and s is a positive number) :

The magnetic field is along the z axis in a Stern-Gerlach experiment.The force it exerts on a magnetic dipole with dipole moment is proportional to:

The Pauli exclusion principle is obeyed by:

When a lithium atom is made from a helium atom by adding a proton (and neutron) to the nucleus and an electron outside, the electron goes into an n = 2, ℓ=0 state rather than an n = 1, ℓ=0 state.This is an indication that electrons:

Electrons are in a two-dimensional square potential energy well with sides of length L.The potential energy is infinite at the sides and zero inside.The single-particle energies are given by where nx and ny are integers.The number of single-particle states with energy 5(h2/8mL2) is:

Electrons are in a two-dimensional square potential energy well with sides of length L.The potential energy is infinite at the sides and zero inside.The single-particle energies are given by , , where n_x and n_y are integers.At most the number of electrons that can have energy 8(h²/8mL²) is:

The quantity s_z is related to the quantum number m_sby:

Five electrons are in a two-dimensional square potential energy well with sides of length L.The potential energy is infinite at the sides and zero inside.The single-particle energies are given by where n_x and n_y are integers.The energy of the ground state of the system is

An electron in an atom is in a state with =3 and m_{11ef01aa_20b1_8f08_bc85_9d99908f7194_TB6585_11} =2.The angle between 11eab079_97ce_6ad3_a51f_1bcb7f4d4386_TB6585_11 and the z axis is:

In the relation µ_z=-m_$\mu$µ_B, the quantity $\mu$ _B is:

Five electrons are in a two-dimensional square potential energy well with sides of length L.The potential energy is infinite at the sides and zero inside.The single-particle energies are given by where n_x and n_y are integers.The energy of the first excited state of the system is:

Electrons are in a two-dimensional square potential energy well with sides of length L.The potential energy is infinite at the sides and zero inside.The single-particle energies are given by where n_x and n_y are integers.The number of single-particle states with energy 5(h²/8mL²) is: