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Questions and Answers
The energy difference between the n = 1 and n = 2 levels for an electron confined to a 1-D box is $1.13 \times 10^{-18} J$.
The energy difference between the n = 1 and n = 2 levels for an electron confined to a 1-D box is $1.13 \times 10^{-18} J$.
True
For a marble confined to a 1-D box of length 0.10 m, the energy difference between n = 1 and n = 2 levels is $5.48 \times 10^{-62} J$.
For a marble confined to a 1-D box of length 0.10 m, the energy difference between n = 1 and n = 2 levels is $5.48 \times 10^{-62} J$.
True
The wavelength corresponding to the spectral transition for an electron is $1.76 \times 10^{-7} m$.
The wavelength corresponding to the spectral transition for an electron is $1.76 \times 10^{-7} m$.
True
For an electron in a 1-D box, the wavelength corresponding to the energy transition is less than the observed distance of any star.
For an electron in a 1-D box, the wavelength corresponding to the energy transition is less than the observed distance of any star.
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The wavefunctions $\psi_1$ and $\psi_2$ described for a particle-in-a-box system are orthogonal.
The wavefunctions $\psi_1$ and $\psi_2$ described for a particle-in-a-box system are orthogonal.
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An electron confined in a box of length $4.0 \times 10^{-10} m$ can have a measurable energy difference of $5.48 \times 10^{-62} J$.
An electron confined in a box of length $4.0 \times 10^{-10} m$ can have a measurable energy difference of $5.48 \times 10^{-62} J$.
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The formula for calculating energy levels in a 1-D box involves the mass of the particle and the length of the box.
The formula for calculating energy levels in a 1-D box involves the mass of the particle and the length of the box.
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The wavelength calculated for an electron transition falls within the infrared region of the electromagnetic spectrum.
The wavelength calculated for an electron transition falls within the infrared region of the electromagnetic spectrum.
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The box length in the conjugated π-electron system is determined solely by the distance between two carbons.
The box length in the conjugated π-electron system is determined solely by the distance between two carbons.
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The empirical parameter p is derived from theoretical calculations rather than experimental results.
The empirical parameter p is derived from theoretical calculations rather than experimental results.
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The integral for the operation of x involves a multiplication of sin functions.
The integral for the operation of x involves a multiplication of sin functions.
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The expression for the box length a considers only the number of carbons in the conjugated system.
The expression for the box length a considers only the number of carbons in the conjugated system.
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The equation x^2 is evaluated over the interval from 0 to a in the context of the provided solution.
The equation x^2 is evaluated over the interval from 0 to a in the context of the provided solution.
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The variable n in the context of formulas generally represents the length of the molecule.
The variable n in the context of formulas generally represents the length of the molecule.
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The formula for R in the solution is defined as the integral of ψ∗ ψ dτ.
The formula for R in the solution is defined as the integral of ψ∗ ψ dτ.
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The term 'nπ' appears in the context of evaluating sin functions in the solution.
The term 'nπ' appears in the context of evaluating sin functions in the solution.
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The operator for p is represented as $\hat{x}^2$.
The operator for p is represented as $\hat{x}^2$.
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The eigenvalue equation is used to calculate the expectation value of $p^2_x$ for a particle in a 1-D box.
The eigenvalue equation is used to calculate the expectation value of $p^2_x$ for a particle in a 1-D box.
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The integral $\int_0^a \sin^2(n\pi x/a)dx$ equals $\frac{a}{2}$.
The integral $\int_0^a \sin^2(n\pi x/a)dx$ equals $\frac{a}{2}$.
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The formula $sin^2\theta = 2sin\theta cos\theta$ is incorrect.
The formula $sin^2\theta = 2sin\theta cos\theta$ is incorrect.
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For the operator $\hat{p}$, the formula includes a term with $\hbar^2$ in the denominator.
For the operator $\hat{p}$, the formula includes a term with $\hbar^2$ in the denominator.
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The term $\psi^*$ represents the complex conjugate of the wave function $\psi$.
The term $\psi^*$ represents the complex conjugate of the wave function $\psi$.
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The expectation value $<p^2_x>$ can only be calculated if the wave function is defined in the region $(0, a)$.
The expectation value $<p^2_x>$ can only be calculated if the wave function is defined in the region $(0, a)$.
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The term $n^2 \pi^2$ appears in the formula for the expectation value $<p^2_x>$ for a particle in a box.
The term $n^2 \pi^2$ appears in the formula for the expectation value $<p^2_x>$ for a particle in a box.
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The general solution to the Schrödinger equation is given by 𝛙(𝐱) = 𝐀 𝐬𝐢𝐧𝐤𝐱 + 𝐁 𝐜𝐨𝐬𝐤𝐱.
The general solution to the Schrödinger equation is given by 𝛙(𝐱) = 𝐀 𝐬𝐢𝐧𝐤𝐱 + 𝐁 𝐜𝐨𝐬𝐤𝐱.
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The value of k is defined as $k = \frac{\sqrt{2mE}}{\hbar}$.
The value of k is defined as $k = \frac{\sqrt{2mE}}{\hbar}$.
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The only solution to the Schrödinger equation when V is infinite is 𝛙 = 1.
The only solution to the Schrödinger equation when V is infinite is 𝛙 = 1.
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According to the given equations, the quantized energy levels can be derived from the equation $E_n = \frac{n^2 \pi^2 \hbar^2}{2ma}$.
According to the given equations, the quantized energy levels can be derived from the equation $E_n = \frac{n^2 \pi^2 \hbar^2}{2ma}$.
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The boundary condition at x = a only affects the coefficient B in the general solution.
The boundary condition at x = a only affects the coefficient B in the general solution.
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The lowest value for the quantum number n is 0.
The lowest value for the quantum number n is 0.
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A particle constrained between x = 0 and x = a has continuous energy values.
A particle constrained between x = 0 and x = a has continuous energy values.
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The equation $\frac{d^2 \Psi}{dx^2} = -k^2 \Psi$ indicates a particle in a box.
The equation $\frac{d^2 \Psi}{dx^2} = -k^2 \Psi$ indicates a particle in a box.
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The energy change from the nth to (n+1)th level decreases with increasing n.
The energy change from the nth to (n+1)th level decreases with increasing n.
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In the particle-in-a-box model, the potential is zero inside the box and infinite outside the box.
In the particle-in-a-box model, the potential is zero inside the box and infinite outside the box.
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The equation for wavelength $\lambda$ in terms of energy $\Delta E$ involves the constants h and c.
The equation for wavelength $\lambda$ in terms of energy $\Delta E$ involves the constants h and c.
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The maximum wavelength $\lambda_{max}$ for polyenylic ions is determined solely by the value of n.
The maximum wavelength $\lambda_{max}$ for polyenylic ions is determined solely by the value of n.
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The time-independent Schrödinger equation is represented as $\hat{H} \Psi(x, y) = E \Psi(x, y)$.
The time-independent Schrödinger equation is represented as $\hat{H} \Psi(x, y) = E \Psi(x, y)$.
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Separation of variables is a technique used to solve the time-independent Schrödinger equation.
Separation of variables is a technique used to solve the time-independent Schrödinger equation.
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The formula for energy levels $E_n$ includes a term $𝑛^2$ multiplied by $h^2$.
The formula for energy levels $E_n$ includes a term $𝑛^2$ multiplied by $h^2$.
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The term $\cos 30^{ ext{o}}$ is included in the expressions for energy levels and wavelength.
The term $\cos 30^{ ext{o}}$ is included in the expressions for energy levels and wavelength.
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The equation $\frac{1}{d^2 \Psi_x} - \frac{1}{d^2 \Psi_y} = 2mE$ shows the energy levels depend on different variables.
The equation $\frac{1}{d^2 \Psi_x} - \frac{1}{d^2 \Psi_y} = 2mE$ shows the energy levels depend on different variables.
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In the normalized wavefunction, if $n_x$ is equal to 3, $n_y$ must also be equal to 3.
In the normalized wavefunction, if $n_x$ is equal to 3, $n_y$ must also be equal to 3.
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The allowed energy levels can be expressed as $E = \frac{h^2 n_x^2 n_y^2}{8mab}$.
The allowed energy levels can be expressed as $E = \frac{h^2 n_x^2 n_y^2}{8mab}$.
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If both sides of equation (37) are equal to a constant, it implies a relationship between $\Psi_x$ and $\Psi_y$.
If both sides of equation (37) are equal to a constant, it implies a relationship between $\Psi_x$ and $\Psi_y$.
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The term $\frac{\hbar^2 d^2 \Psi_y}{2m d y^2} = E_y \Psi_y$ states that the energy $E_y$ is proportional to the second derivative of the wavefunction $\Psi_y$.
The term $\frac{\hbar^2 d^2 \Psi_y}{2m d y^2} = E_y \Psi_y$ states that the energy $E_y$ is proportional to the second derivative of the wavefunction $\Psi_y$.
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The variable 'a' in the equations corresponds to the width of the box in the normalized wavefunction.
The variable 'a' in the equations corresponds to the width of the box in the normalized wavefunction.
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In the equations provided, $E = E_x + E_y$ means the total energy is the sum of the energy contributions from both dimensions.
In the equations provided, $E = E_x + E_y$ means the total energy is the sum of the energy contributions from both dimensions.
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If the sides of the box are not equal, it implies that $E_x$ and $E_y$ will have the same values.
If the sides of the box are not equal, it implies that $E_x$ and $E_y$ will have the same values.
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Equation (38) indicates that the kinetic energy term is invariant under one variable's transformation.
Equation (38) indicates that the kinetic energy term is invariant under one variable's transformation.
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The normalization condition ensures that the wavefunctions are defined over the dimensions of the box correctly.
The normalization condition ensures that the wavefunctions are defined over the dimensions of the box correctly.
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Study Notes
KFT431 Physical Chemistry III: Quantum Chemistry (Part II)
- Course covers Quantum mechanics for simple systems, including the free particle, particle-in-a-box (1-D, 2-D, and 3D), harmonic oscillator, and rigid rotor.
-
Free Particle:
- Total energy is the kinetic energy.
- Energy is not quantized.
- Probability of finding the particle at any point along the x-axis is equal.
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Particle-in-a-Box (1-D):
- Potential energy is zero for 0 < x < a and infinite outside this region.
- Energy levels are quantized: En = n2h2/8ma2, where n=1,2,3...
- Wavefunction ψn(x) = √(2/a)sin(nπx/a).
-
Particle-in-a-Box (Multiple dimensions):
-Energy levels are quantized by Enx,ny = (h2/8m)(nx2/a2 + ny2/b2) for a 2 dimensional box. For a 3 dimensional box, E = h2/8m(nx2/a2 + ny2/b2 + nz2/c2) -The solutions to these equations in multiple dimensions are similar to the 1-dimenstional case: ψnx, ny (x,y) = √(2/a)(2/b)sin(nx πx/a)sin(ny πy/b) -
Harmonic Oscillator:
- The allowed energy levels are equally spaced.
- Zero-point energy exists.
- Ev = (v + 1/2)hν, where v is the vibrational quantum number.
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Rigid Rotor:
- Represents diatomic molecules rotating with constant distance btwn the atoms.
- Energy (E) = J (J+1) h2 / 2I, where J is rotational quantum number.
- The rotational levels are (2J +1) fold degenerate.
Quantum Mechanical Operators
- Operators for quantum mechanical quantities, and how they relate to classical versions.
Examples of Calculations
- Calculations of energy differences, transition wavelengths.
- Probability of finding a particle within a given range.
- Expected values for different quantum mechanical operators .
Degeneracy
- When energy levels correspond to more than one state, it's called degenerate.
Application of Particle in a Box
- Calculations of electronic spectra of conjugated π-electron systems provide insight into molecular structures.
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Description
Explore the principles of quantum mechanics in simple systems, including the free particle, particle-in-a-box in one, two, and three dimensions, harmonic oscillator, and rigid rotor. This quiz will test your understanding of energy quantization and wavefunctions. Dive deep into the fascinating world of quantum chemistry!
