Homework Answers
The wavefunction given is a solution of the schrodinger equation, if it it satisfies the equation. That is LHS= RHS. Please see below for the calculations:
Note that if the wavefunction satisfies the schrodinger equation, means the the eigenvalue is E or energy.
For c) The graphs are drawn
below.
For d) The vibrational frequency formula is used. Note that since it is given in per cm units, we will use c= speed of light in cm/s unit . Also, mass should be converted to SI unit of kg. Using the relation 1 amu = 1.67 x 10-27 kg the conversion is done.
On solving the value of k = 492.99 Nm-1 or 493 Nm-1
Note that kg.m.s-2 = 1 N
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a) Show that the wave function y(x) = N exp( – x²/(2a?)) with a? = ()...
Potential energy function, V(x) = (1/2)mw2x2 Assuming the time-independent Schrödinger equation, show that the following wave...
Potential energy function,
V(x) = (1/2)mw2x2
Assuming the time-independent Schrödinger equation, show that the following wave functions are solutions describing the one-dimensional harmonic behaviour of a particle of mass m, where ?2-h/v/mK, and where co and ci are constants. Calculate the energies of the particle when it is in wave-functions ?0(x) and V1 (z) What is the general expression for the allowed energies En, corresponding to wave- functions Un(x), of this one-dimensional quantum oscillator? 6 the states corresponding to the...
= μ = 0.5 This problem deals with the vibrational motion of the H2 molecule (reduced mass- amu). The Hamiltonian for this system is: h2 d 1, e2ndxī + 2kx2. 5 pts] By direct substitution of the wa...
= μ = 0.5 This problem deals with the vibrational motion of the H2 molecule (reduced mass- amu). The Hamiltonian for this system is: h2 d 1, e2ndxī + 2kx2. 5 pts] By direct substitution of the wavefunction labelled by the quantum number v, Where k is a constant related to the bond strength. V.(x), in the Schrödinger Equation, show that the wavefunction Ψ(x) = Noe- )' where α = ( corresponds to the ground vibrational state of H2 having...
h2 4. In a region of the x-axis, a particle has a wave function given by...
h2 4. In a region of the x-axis, a particle has a wave function given by y(x) = Ae-*4722° and energy where L is some length. (a) Find the potential energy as a function of x, and sketch V (x) versus x. (b) What is the classical potential (or corresponding force function) that has this dependence? (c) Find the kinetic energy as a function of x. (d) Show that x = L is the classical turning point (i.e. the place...
A one-particle two-dimensional harmonic oscillator has the potential energy function V=V(x,y)=k/2(x2+y2). write the t...
A one-particle two-dimensional harmonic oscillator has the potential energy function V=V(x,y)=k/2(x2+y2). write the time-independent SchrÖdinger equation for the system and the energy eigenvalues. Define clearly the symbols you used.
4&5 only thnkyouu :) 3. The force constant for 119F molecule is 966 N/m. a) Calculate...
4&5 only
thnkyouu :)
3. The force constant for 119F molecule is 966 N/m. a) Calculate the zero-point vibrational energy using a harmonic oscillator potential. b) Calculate the frequency of light needed to excite this molecule from the ground state to the first excited state. 4. Is 41(x) = *xe 2 an eigenfunction for the kinetic energy operator? Is it an eigenfunction for potential energy operator? 5. HCI molecule can be described by the Morse potential with De = 7.41...
Determine the energy of the particle if the proposed wave function satisfies the Schrödinger equation for x < 0.
A fellow student proposes that a possible wave function for a free particle with mass \(m\) (one for which the potential-energy function \(U(x)\) is zero ) is$$ \psi(x)=\left\{\begin{array}{ll} e^{-k x}, & x \geq 0 \\ e^{+\kappa x}, & x<0 \end{array}\right. $$where \(\kappa\) is a positive constant. (a) Graph this proposed wave function.(b) Determine the energy of the particle if the proposed wave function satisfies the Schrödinger equation for \(x<\)0.(c) Show that the proposed wave function also satisfies the Schrödinger equation...
The most general wave function of a particle in the simple harmonic oscillator potential is: V(x,...
The most general wave function of a particle in the simple harmonic oscillator potential is: V(x, t) = (x)e-1st/ where and E, are the harmonic oscillator's stationary states and their corresponding energies. (a) Show that the expectation value of position is (hint: use the results of Problem 4): (v) = A cos (wt - ) where the real constants A and o are given by: 1 2 Ae-id-1 " Entichtin Interpret this result, comparing it with the motion of a...
Dont do Part A. A localized electron at rest has a wave function ψ(x)=A exp( a22.2)...
Dont do Part A.
A localized electron at rest has a wave function ψ(x)=A exp( a22.2) with a=0.5/nm. (a) Use the results from class to quote it's space and momentum uncertainty (b) Use the static Schrödinger equation to calculate the pertinent potential, U(x).
1000 diatomic molecules with vibrational state described by N molecules in the ground vibrational...
1000 diatomic molecules with vibrational state described by N molecules in the ground vibrational state O molecules in the lowest potential Total Energy Potential Energy state M molecules in an excited vibrational states P molecules in an excited potential energy states Schematic of Energy Eigenfunctions and the 1. Consider a sample of 1000 identically prepared diatomic molecules, each of which can be Potential Energy function of the Harmonic Oscillator described by the ground state of the Harmonic oscillator: Ψ-ψ。. If...
2. The vibrational frequency of gaseous N O is 1904 cm. Assume this molecule is a harmonic oscillator 2.1 What is the energy of the electromagnetic wave corresponding to this vibrational frequenc...
2. The vibrational frequency of gaseous N O is 1904 cm. Assume this molecule is a harmonic oscillator 2.1 What is the energy of the electromagnetic wave corresponding to this vibrational frequency? 2.2 Calculate the force constant of "NO 2.3 Calculate the vibrational frequency of gaseous N O. The isotopic effect does not change the force constant of the harmonic oscillator. 2.4 When "N'O is bound to hemoglobin A (Hb or Hgb, the iron-containing oxygen-transport metalloprotein in the red blood...
Potential energy function,
V(x) = (1/2)mw2x2
Assuming the time-independent Schrödinger equation, show that the following wave functions are solutions describing the one-dimensional harmonic behaviour of a particle of mass m, where ?2-h/v/mK, and where co and ci are constants. Calculate the energies of the particle when it is in wave-functions ?0(x) and V1 (z) What is the general expression for the allowed energies En, corresponding to wave- functions Un(x), of this one-dimensional quantum oscillator? 6 the states corresponding to the...
= μ = 0.5 This problem deals with the vibrational motion of the H2 molecule (reduced mass- amu). The Hamiltonian for this system is: h2 d 1, e2ndxī + 2kx2. 5 pts] By direct substitution of the wavefunction labelled by the quantum number v, Where k is a constant related to the bond strength. V.(x), in the Schrödinger Equation, show that the wavefunction Ψ(x) = Noe- )' where α = ( corresponds to the ground vibrational state of H2 having...
h2 4. In a region of the x-axis, a particle has a wave function given by y(x) = Ae-*4722° and energy where L is some length. (a) Find the potential energy as a function of x, and sketch V (x) versus x. (b) What is the classical potential (or corresponding force function) that has this dependence? (c) Find the kinetic energy as a function of x. (d) Show that x = L is the classical turning point (i.e. the place...
4&5 only
thnkyouu :)
3. The force constant for 119F molecule is 966 N/m. a) Calculate the zero-point vibrational energy using a harmonic oscillator potential. b) Calculate the frequency of light needed to excite this molecule from the ground state to the first excited state. 4. Is 41(x) = *xe 2 an eigenfunction for the kinetic energy operator? Is it an eigenfunction for potential energy operator? 5. HCI molecule can be described by the Morse potential with De = 7.41...
The most general wave function of a particle in the simple harmonic oscillator potential is: V(x, t) = (x)e-1st/ where and E, are the harmonic oscillator's stationary states and their corresponding energies. (a) Show that the expectation value of position is (hint: use the results of Problem 4): (v) = A cos (wt - ) where the real constants A and o are given by: 1 2 Ae-id-1 " Entichtin Interpret this result, comparing it with the motion of a...
Dont do Part A.
A localized electron at rest has a wave function ψ(x)=A exp( a22.2) with a=0.5/nm. (a) Use the results from class to quote it's space and momentum uncertainty (b) Use the static Schrödinger equation to calculate the pertinent potential, U(x).
1000 diatomic molecules with vibrational state described by N molecules in the ground vibrational state O molecules in the lowest potential Total Energy Potential Energy state M molecules in an excited vibrational states P molecules in an excited potential energy states Schematic of Energy Eigenfunctions and the 1. Consider a sample of 1000 identically prepared diatomic molecules, each of which can be Potential Energy function of the Harmonic Oscillator described by the ground state of the Harmonic oscillator: Ψ-ψ。. If...
2. The vibrational frequency of gaseous N O is 1904 cm. Assume this molecule is a harmonic oscillator 2.1 What is the energy of the electromagnetic wave corresponding to this vibrational frequency? 2.2 Calculate the force constant of "NO 2.3 Calculate the vibrational frequency of gaseous N O. The isotopic effect does not change the force constant of the harmonic oscillator. 2.4 When "N'O is bound to hemoglobin A (Hb or Hgb, the iron-containing oxygen-transport metalloprotein in the red blood...
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