Other Principles of Quantum Mechanics - AP Physics 2
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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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