Principles of Quantum Mechanics - AP Physics 2
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Determine the frequency of orange light, 
.
Determine the frequency of orange light,
Using
Converting 
to 
and plugging in values
Using
Converting
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Determine the photon energy of orange light, 
.
Determine the photon energy of orange light,
Using
is the photon energy
is Plank's constant, 
is the speed of light, 
is the wavelength
Converting 
to 
and plugging in values:
Using
Converting
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A beam of light is traveling at approximately 
in a very dense medium. Given that the frequency of the light beam is 
, what is the wavelength of the light beam? Suppose that the speed of light in a vacuum is 
.
A beam of light is traveling at approximately
The wavelength of the light beam doesn't depend on the speed through the medium. As such, as can solve for wavelength 
by simply solving:
Where 
is frequency and 
is the speed of light in a vacuum.
The wavelength of the light beam doesn't depend on the speed through the medium. As such, as can solve for wavelength
Where
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How much energy is contained in a particle that has a mass of 
?
How much energy is contained in a particle that has a mass of
This is an example of one of Einstein's greatest ideas: the relation between the mass of an object/particle, and the energy contained by the mass. This is given as
In order to calculate the energy in our particle, we must make sure that the mass is in units of 
.
Now we can plug in numbers to our equation and solve for energy.
This is an example of one of Einstein's greatest ideas: the relation between the mass of an object/particle, and the energy contained by the mass. This is given as
In order to calculate the energy in our particle, we must make sure that the mass is in units of
Now we can plug in numbers to our equation and solve for energy.
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Suppose that the mass of a neutral Uranium atom is measured and found to be 
. However, after adding up the mass of all constituent protons, neutrons, and electrons, the predicted mass of a Uranium atom is expected to be equal to 
. Based on this information, what is the nuclear binding energy of a uranium atom?
Suppose that the mass of a neutral Uranium atom is measured and found to be
In this question, we're presented with information concerning the mass of a uranium atom. We're told two values: the mass of a uranium atom as measured, and the predicted mass of a uranium atom. We're then asked to determine the nuclear binding energy for uranium.
In order to solve this question, we have to realize the significance of the discrepancy between the observed and predicted mass of uranium. The predicted mass is calculated by adding up the individual masses of each constituent proton, electron, and neutron. However, the reason why the measured mass is less than the predicted mass is due to energy-mass equivalence. When the constituent protons and neutrons come together to form the nucleus, some of their mass is converted into energy, and it is this energy which holds these constituent nucleons together. Because some of the mass is converted into energy, the observed mass is less than what we would predict.
Now that we understand why there is a discrepancy between observed and predicted mass, we can calculate the nuclear binding energy by using Einstein's famous equation.
This equation states that the nuclear binding energy is equal to the difference between observed and predicted mass, multiplied by the speed of light squared. So to solve for energy, we can plug in the values given to us.
In this question, we're presented with information concerning the mass of a uranium atom. We're told two values: the mass of a uranium atom as measured, and the predicted mass of a uranium atom. We're then asked to determine the nuclear binding energy for uranium.
In order to solve this question, we have to realize the significance of the discrepancy between the observed and predicted mass of uranium. The predicted mass is calculated by adding up the individual masses of each constituent proton, electron, and neutron. However, the reason why the measured mass is less than the predicted mass is due to energy-mass equivalence. When the constituent protons and neutrons come together to form the nucleus, some of their mass is converted into energy, and it is this energy which holds these constituent nucleons together. Because some of the mass is converted into energy, the observed mass is less than what we would predict.
Now that we understand why there is a discrepancy between observed and predicted mass, we can calculate the nuclear binding energy by using Einstein's famous equation.
This equation states that the nuclear binding energy is equal to the difference between observed and predicted mass, multiplied by the speed of light squared. So to solve for energy, we can plug in the values given to us.
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Two grams of helium are completely converted into energy and used to power a 
man. If all of this energy is converted into kinetic energy of the man, how fast will he move?
Two grams of helium are completely converted into energy and used to power a
The energy from the two grams of helium can be found using 
This energy can then be equated to the man's kinetic energy, which can then be used to find the man's velocity.
The energy from the two grams of helium can be found using
This energy can then be equated to the man's kinetic energy, which can then be used to find the man's velocity.
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If the combination of protons and neutrons in an atom's nucleus results in a mass defect of 
, what is the binding energy for this atom?
If the combination of protons and neutrons in an atom's nucleus results in a mass defect of
In this question, we're given the mass defect of an atom's nucleus and are asked to find the binding energy for this atom.
To begin with, it's important to understand that when protons and neutrons come to be held together within the nucleus of an atom, there is a tremendously powerful force holding them together. This incredibly large force accounts for the mass defect. In other words, the total mass of the nucleus is smaller than the sum of the individual masses of the protons and neutrons that make up that nucleus, and this is due to the strong force.
Einstein's mass-energy equivalence explains the observable mass defect; the mass lost is converted into an enormous amount of energy according to the following equation.
But first, we'll need to convert the mass given to us in the question stem into grams.
Furthermore, because we know the value for the speed of light, we can use this information to solve for the binding energy.
In this question, we're given the mass defect of an atom's nucleus and are asked to find the binding energy for this atom.
To begin with, it's important to understand that when protons and neutrons come to be held together within the nucleus of an atom, there is a tremendously powerful force holding them together. This incredibly large force accounts for the mass defect. In other words, the total mass of the nucleus is smaller than the sum of the individual masses of the protons and neutrons that make up that nucleus, and this is due to the strong force.
Einstein's mass-energy equivalence explains the observable mass defect; the mass lost is converted into an enormous amount of energy according to the following equation.
But first, we'll need to convert the mass given to us in the question stem into grams.
Furthermore, because we know the value for the speed of light, we can use this information to solve for the binding energy.
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A proton is confined to a one-dimensional box of length 
. It has an energy equal to that of a photon with a wavelength of 
. What excited state is the proton in? (Remember, the first excited state is where 
since the ground state is 
).
A proton is confined to a one-dimensional box of length
The energy of the quantum system in the 
state is given by
where 
is Planck's constant, 
is the mass of the proton and 
is the length of the box. The energy of a photon is given by
where 
is the frequency, 
is the speed of light and 
is the wavelength. Setting these equal we can solve for 
,
Since the ground state is 
, the proton must be in the second excited state.
The energy of the quantum system in the
where
where
Since the ground state is
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The expectation value 
of a particle in a quantum system tells us what about the particle?
The expectation value
From a statistical standpoint, the expectation value of the position, 
, can only tell us the most probable location of the particle. A central idea in quantum mechanics is that we can never really know exactly where a particle is as a function of time, but rather where we are most likely to find the particle if we choose to observe it.
From a statistical standpoint, the expectation value of the position,
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If a particle in a quantum system is bounded, this means what about the calculated particle energy?
If a particle in a quantum system is bounded, this means what about the calculated particle energy?
A key characteristic of bound-state systems is the quantization of the energy into discrete energy levels. The energy can be negative or positive, but the particle can only have an energy that corresponds to one of these energy levels, nothing other. An example is the energy of a hydrogen atom, where the energy levels are given by:
These discrete numbers come from the energy of an electron, which is the fundamental charge.
A key characteristic of bound-state systems is the quantization of the energy into discrete energy levels. The energy can be negative or positive, but the particle can only have an energy that corresponds to one of these energy levels, nothing other. An example is the energy of a hydrogen atom, where the energy levels are given by:
These discrete numbers come from the energy of an electron, which is the fundamental charge.
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The square modulus of the wavefunction, given as 
, contains what information about a particle in a quantum system?
The square modulus of the wavefunction, given as
By definition, 
represents the probability distribution of a particle in a quantum system as a function of time. It is used to calculate the expectation value of other observables, such as position, momentum, current, angular momentum, just to name a few.
By definition,
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A physicist is trying to determine the identity of a metal by experimentally finding its work function and then comparing the experimental value to a list of known values. If the physicist shines light with a frequency of 
on the metal and records an electron speed of 
, what is the work function of the metal?
A physicist is trying to determine the identity of a metal by experimentally finding its work function and then comparing the experimental value to a list of known values. If the physicist shines light with a frequency of
To determine the work function of the metal, we can begin by calculating the kinetic energy of the electron from its speed.
Then, convert joules into electron volts:
Next, we'll calculate the energy of the light by using its frequency according to the equation:
Then, we can convert this energy into electron volts:
And finally, we can use the equation for work function:
To determine the work function of the metal, we can begin by calculating the kinetic energy of the electron from its speed.
Then, convert joules into electron volts:
Next, we'll calculate the energy of the light by using its frequency according to the equation:
Then, we can convert this energy into electron volts:
And finally, we can use the equation for work function:
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An atomic transition takes place from 
to the ground state 
. What is the wavelength of the photon this transition will emit?
An atomic transition takes place from
The energy difference between the second excited state 
and the ground state 
will be equal to the photon energy since energy is conserved.
Since the energy difference is in electron volts, they need to be converted to Joules. Another convenient formula is to notice that since 
and 
are constants, they can be converted to electron volts and you can use
The energy difference between the second excited state
Since the energy difference is in electron volts, they need to be converted to Joules. Another convenient formula is to notice that since
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It is possible that electrons can be ejected from a metal plate by light. Which of the following properties of light accounts for this phenomena?
It is possible that electrons can be ejected from a metal plate by light. Which of the following properties of light accounts for this phenomena?
The ejection of electrons from a metal is known as the photoelectric effect. In fact, it was a key discovery that led to the acceptance of the particle nature of light, because the wave nature of light could not account for this finding.
The key that allowed scientists to find this out was that the speed of electrons ejected from the metal was dependent on the color of the light. The intensity of light had no effect on the speed at which electrons were ejected.
As a part of developing this theory, scientists referred to individual discrete particles of light as photons. In order to eject electrons from the metal, individual photons needed to have a sufficient amount of energy. The intensity of light (with a greater amplitude of its waves) represents a case in which photons are striking the metal at a greater rate. But, the individual energy of the photons could be too low in such a situation, hence why intensity is not the key property here. Instead, a higher frequency of light allows each individual photon to have greater energy - enough to knock the metal's electrons off. Furthermore, as the frequency of the light increases, the energy of the individual photons increases and the kinetic energy of the ejected electrons increases as well. The expression for the energy of a photon is shown as follows.
The ejection of electrons from a metal is known as the photoelectric effect. In fact, it was a key discovery that led to the acceptance of the particle nature of light, because the wave nature of light could not account for this finding.
The key that allowed scientists to find this out was that the speed of electrons ejected from the metal was dependent on the color of the light. The intensity of light had no effect on the speed at which electrons were ejected.
As a part of developing this theory, scientists referred to individual discrete particles of light as photons. In order to eject electrons from the metal, individual photons needed to have a sufficient amount of energy. The intensity of light (with a greater amplitude of its waves) represents a case in which photons are striking the metal at a greater rate. But, the individual energy of the photons could be too low in such a situation, hence why intensity is not the key property here. Instead, a higher frequency of light allows each individual photon to have greater energy - enough to knock the metal's electrons off. Furthermore, as the frequency of the light increases, the energy of the individual photons increases and the kinetic energy of the ejected electrons increases as well. The expression for the energy of a photon is shown as follows.
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A student on the Earth observes a spaceship moving at a considerable fraction of the speed of light. As a result of the spaceship's motion, the student observes that the clocks aboard the spaceship are running slower than when the spaceship was on the Earth. When a student aboard the spaceship observes the student's clock on the Earth, what effect will he notice on the rate of the Earth-bound clock?
A student on the Earth observes a spaceship moving at a considerable fraction of the speed of light. As a result of the spaceship's motion, the student observes that the clocks aboard the spaceship are running slower than when the spaceship was on the Earth. When a student aboard the spaceship observes the student's clock on the Earth, what effect will he notice on the rate of the Earth-bound clock?
For the observer on the spaceship, the Earth appears to be moving, and special relativity tells us that any observer moving at constant speed can consider him/herself stopped. Therefore, the clock on the Earth experiences the same slowing from the spaceship's viewpoint as the spaceship's clock from the Earth viewpoint. It's a little bit like looking at someone from far away. They look small due to the distance, but you do not look big to them, you also look smaller.
For the observer on the spaceship, the Earth appears to be moving, and special relativity tells us that any observer moving at constant speed can consider him/herself stopped. Therefore, the clock on the Earth experiences the same slowing from the spaceship's viewpoint as the spaceship's clock from the Earth viewpoint. It's a little bit like looking at someone from far away. They look small due to the distance, but you do not look big to them, you also look smaller.
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Determine the observed length of a 
rod traveling along it's long axis at 
in relation to an observer.
Determine the observed length of a
Using the formula for length contraction:
Where 
is the rest length,
is the velocity of the object
is the speed of light
is the observed length
Plugging in values
Using the formula for length contraction:
Where
Plugging in values
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Determine the observed length of a 
rod traveling along it's long axis at 
in relation to an observer.
Determine the observed length of a
Use the following equation:
Where is the rest length,
is the velocity of the object
is the speed of light
is the observed length
Plugging in values
Use the following equation:
Where
Plugging in values
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A 
long rod is traveling at 
in relationship to an observer along it's long axis. Determine the observed length.
A
Using
Plugging in values
Using
Plugging in values
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A 
tall rocket is traveling at 
in relationship to an observer along it's long axis. Determine the observed length.
A
Using
Plugging in values
Using
Plugging in values
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