Chapter 8B

The hamiltonian operator for the harmonic oscillator is ___ - -p^2/2m*delta^2/x2 + 1/2 kx^2 - -p^2/2m*delta^2/x2 - 1/2 kx^2 - -p^2/2m*delta^2/x2 + 1/2 kx -None of the other choices is correct - -p^2/2m*delta^2/x2

-p^2/2m*delta^2/x2 + 1/2 kx^2

What is the mean displacement of a harmonic oscillator? -1 -0 -The other choices are ALL incorrect. -hbarw/2 -hbarw

0

A mass m = 2.0 kg is attached to a spring having a force constant k = 290 N/m as in the figure. The mass is displaced from its equilibrium position and released. Its frequency of oscillation (in Hz) is approximately -0.01 -1.9 -0.50 -12 -0.08

1.9

The potential energy of quantum harmonic oscillator is ____ -(-kx) where k is the hooke's law constant -1/2 kx^2, where k is considered to be the spring constant in Hooke's law -(-kx), where k is a constant but not hooke's law constant -1/2 kx(2 where k is a constant not necessarily the hooke's law constant -The potential energy is constant throughout

1/2 kx^2, where k is considered to be the spring constant in Hooke's law

The lowest possible energy for simple harmonic oscillator is E= _____ -None of the other choices is correct -3/2hbarw -1/2hbarw -5/2hbarw -2/1hbarw

1/2hbarw

The potential energy of a harmonic oscillator is____ - (-kx) -delta2/deltax2 -1/2kx^2 -mgh -p^2/2m

1/2kx^2

A graph of position versus time for an object oscillating at the free end of a horizontal spring is shown on the right. A point or points at which the object has positive velocity and zero acceleration is(are) -B or D -A -B -D

A

A mass on a spring undergoes simple harmonic motion. The maximum displacement from the equilibrium is called? -Speed -Wavelength -Frequency -Amplitude -Period

Amplitude

Which of the following statements is TRUE regarding the quantum harmonic oscillator? -The highest probability of finding the particle is at the center of the potential energy, that is, at 2 = 0. -The other statements are ALL correct. -As quantum number increases, the probability of finding the particle at the center of the potential energy well decreases and the probability of being at the sides of the potential well increases. -There is always a node at (x = 0), at the center of the potential well. -As quantum number increases, the probability of finding the particle at the center of the potential energy well increases and the probability of being at the sides of the potential well decreases

As quantum number increases, the probability of finding the particle at the center of the potential energy well decreases and the probability of being at the sides of the potential well increases

The force acting on a pendulum is proportional to -Acceleration -Velocity -Displacement -Time

Displacement

What is the zero-point energy of a quantum harmonic oscillator? _E1 -The other options are ALL incorrect. -E0 -Zero -1/2h

E0

In a periodic process, the number of cycles per unit of time is called? -Amplitude -Frequency -Speed -Wavelength -Period

Frequency

The solution wavefunction, v(x), to the quantum harmonic oscillator, is of the form v(x) = NvHv(y) e-y22 where y=xa and a=(h2mkf)14 Which part of the equation represents a polynomial in x that is a solution to a basic second order differential equation? -e-y2/2 -Hv(y) -The other choices are correct -Nv

Hv(y)

Which of the following statements is NOT TRUE regarding nodes of harmonic oscillator? -Number of nodes is equal to (v-1) -Number of nodes in 1(x) is 1 -Number of nodes is equal to v -The number of nodes depend on the quantum number v -Nodes represent points where there is a zero possibility of finding the particle at those points

Number of nodes is equal to (v-1)

The solution wavefunction, v(x), to the quantum harmonic oscillator, is of the form v(x) = NvHv(y) e-y22 where y=xa and a=(h2mkf)14Which part of the equation represents the factor that is computed to make sure that the total square probability is equal to 1? -e-y2/2 -Hv(y) -The other choices are correct -Nv

Nv

The solution wavefunction, v(x), to the quantum harmonic oscillator, is of the form v(x) = NvHv(y) e-y22 where y=xa and a=(h2mkf)14.Which of the following statement(s) is NOT true regarding the equation? -Hv represents the Hermite polynomials. -e^-y2/2 is a bell-shaped Gaussian function that Will ensure the equation is bounded -The other choices are ALL correct. -Nv represents a quantum number.

Nv represents a quantum number

A mass on a spring undergoes simple harmonic motion. The maximum displacement from the equilibrium is called? -Wavelength -Frequency -Speed -Period -Amplitude

Period

Which of the following statements is/are false regarding simple harmonic motion? -The other choices are ALL correct. -The wavefunctions of a harmonic oscillator are products of a Hermit polynomial and a Gaussian function. -The simple harmonic oscillator has a zero-point energy. -The kinetic energy of a harmonic oscillator is a parabolic function of the displacement from equilibrium. -The difference between the energy level to the next higher or next lower energy level is constant.

The kinetic energy of a harmonic oscillator is a parabolic function of the displacement from equilibrium

Which of the following statements are NOT TRUE regarding the energy levels of a quantum harmonic oscillator? -The energy levels of a harmonic oscillator are evenly spaced. -The space in between an energy level and the next highest is always hbarw -The other choices are ALL correct. -The lowest energy level is when v=1.

The lowest energy level is when v=1.

What can be said of the mean displacement of the particle in quantum harmonic oscillator? -The mean displacement is at the origin (×=0) for antisymmetric wave functions. -Impossible to generalize, you have to compute everytime -The mean displacement is always at the origin, X=0. -The mean displacement is at the origin (×=0) for symmetrical wavefunctions.

The mean displacement is always at the origin, X=0

For a quantum harmonic oscillator system, for the wavefunction psi 20(x)What can be said of the nodes? -There will be 19 nodes in that wavefunction. -A node can be found at x=0. -There will be 20 nodes in that wavefunction. -It is impossible to say anything about the nodes.

There will be 20 nodes in that wavefunction

Which of the following statements is not true regarding Hermite polynomials _The Hermite polynomial satisfy the recursion relation Hv+1- 2y+2vHv-1=0 -The Hermite polynomials are solutions to a second-order differential equation, H''v-2yH'v+ 2vHv=0 -The other statements are ALL correct. -These equations are the full solution to the quantum harmonic oscillator problem.

These equations are the full solution to the quantum harmonic oscillator problem.

Where is the probability of finding the harmonic oscillator in classically forbidden states most significant? -at the ground vibrational state, when v=1 -at the ground vibrational state, when v=0 -The other choices are ALL incorrect. -at large numbers of v, that is, as v-> positive infinite

at the ground vibrational state, when v=0

The solution wavefunction, v(x), to the quantum harmonic oscillator, is of the formv(x) = NvHv(y) e-y22 where y=xa and a=(h2mkf)14 Which part of the equation represents a bell-shaped Gaussian function, placed in the solution to ensure that the wavefunction is bounded? -e-y2/2 -Hv(y) -The other choices are correct -Nv

e-y2/2

What is the relationship between h and hbar? -The other choices are ALL incorrect. -h=hbar/2pi -hbar=2pih -h=2pihbar -hhbar=2pi

h=2pihbar

The symbol, w, in the energy level equation of a quantum harmonic oscillator, is -inversely proportional to the spring constant. -inversely proportional to the square of the mass of the particle. -proportional to the mass of the particle -inversely proportional to the square root of the mass of the particle.

inversely proportional to the square root of the mass of the particle

What is the formula for the reduced mass for two particles with masses m1 and m2? -The geometric average of two masses, sq.root. m1m2 -m1m2/m1+m2 -m1+ m2/m1m2 -m1+m2 -The average of the two masses, 1/2(m1+m2)

m1m2/m1+m2

If two elements X and Y have the mass of X much, much greater than Y, that is, mX>> mY, the effective mass of the two, , may be estimated as -mx/my -mX -Average of the two masses -Difference of the two masses -mY

mY

In describing the classical harmonic motion, which of the following assumptions are FALSE? -the body experiences the resistance of a spring. -the force experienced by the body is governed by Hooke's law. -potential energy is constant at all instances. -there are no friction experienced by the body. -the other choices are ALL assumed to be correct.

potential energy is constant at all instances

Which of the following wavefunctions of the quantum harmonic oscillator is/ are anti-symmetrical about the origin? -The other choices are ALL NOT correct. -psi 0(x) -psi 1(x) -psi 2(x)

psi 1(x)

Which of the following wavefunctions of the quantum oscillator is/are symmetrical about origin -The other choices are all correct -psi 2(x) -psi 3(x) -psi 1(x)

psi 2(x)

The rough illustration below for the wavefunction (the blue curve) represents which particular wavefunction? -psi5(x) -psi7(x) -psi4(x) -psi0(x) -psi3(x) -psi6(x)

psi4(x)

he solution wavefunction to the quantum harmonic oscillator is expressed as v(x). The full form of the equation is v(x) = NvHv(y) e-y22 where y=xa and a=(h2mkf)14 What can be said of v in the equation? -the other choices are all correct. -it can be zero -it cannot be negative. -it represents a quantum number.

the other choices are all correct.

The energy eigenvalue, Ev=, of the simple harmonic oscillator is -hbarw -(v+1/2) hbarw, where w =(kf/m)^1/2 -(v+12) hw, where w =(kf/m)^1/2 -nhbarv -hbarv

v+1/2) hbarw, where w =(kf/m)^1/2

Which of the following is NOT a solution of the differential equation ''(x) + k2y(x) =0? -y(x)=sin (kx-a) where a is constant -y(x)=sin (kx) -y(x)=e^-ikx -y(x)=cos (kx) -y(x)=e^-kx2

y(x)=e^-kx2