- #1
fox26
- 40
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This problem bothered me many years ago when I was taking a university course in
quantum mechanics, but I assumed it was due to an error on my part which I, however,
couldn’t locate, and I didn’t ask my course instructor about it. Recently, when
researching a q.m. question on the Internet, I rediscovered it: Both in my textbooks years
ago and on the Internet now, (non-normalizable) plane-wave solutions to the
non-Relativistic, 1-dimensional, but easily extendable to 3 dimensions, Schrodinger
wave equation for any free particle PRTCL with non-zero rest mass (which always
moves at less than the speed of light), such as an electron, are given as proportional to
Ψ = exp(i[kx-ωt]), where k = 2π/λ, λ being Ψ’s wavelength, ω = 2πf, where f is Ψ’s
frequency, i is the imaginary constant, x is the length coordinate, and t is time. The
problem arises from the identification of h/λ with the classical momentum p = ms of
PRTCL, and hf with its classical kinetic energy KE = p2/2m = ms2/2, where m is PRTCL’s
mass, s is its speed, and h is Planck’s Constant. Then Ψ = exp(2πi[px -KEt]/h) = exp(2π
i[msx - ms2 t/2]/h), and if t increases one unit, the term it is in increases by ms2 /2, so for
Ψ to have the same phase as before, x must increase by ms2/2ms = s/2 units, so the
phase velocity (speed) of Ψ is s/2. This seems to be at variance with the speed of
PRTCL's being s, but it might be objected that Ψ is not a physically realistic wavefunction,
being non-normalizable. However, Ψ can be modified by smoothly limiting it to be zero
outside of some large but finite region R, but to be the same as before inside of some
region R’ inside but nearly as big as R, so that the modified wave function Ψ' is
normalizable, and so that Ψ' is the wave function of some free particle mPRTCL with
only slightly uncertain momentum and speed, with momentum expectation value p and
speed expectation value s, while the phase velocity of the main Fourier component of Ψ′
and the speed of change of its position expectation value is s/2.
The cause of this discrepancy seems to be that KE = hf, which Louis de Broglie, the
inventor of the idea of the wave function of a particle, is supposed to have assumed to
be true for all particles, is true for photons, which move with the speed of light, and all of
whose energy E = mc2 is kinetic energy, none being their rest mass equivalent energy,
so their KE = mc2 = ms2 (E = hf for a photon was Planck’s quantum hypothesis, which began
quantum mechanics, made to avoid the Ultraviolet Catastrophe of then-current
theory [maybe the UC wasn’t implied by then-current-theory, and I’ll post a thread about
that]), but KE = hf, where f is the true frequency of the wave function of a particle in a
momentum eigenstate (i.e., the value of f in the equation above which is necessary to
make the phase velocity, or the speed of the movement of the expectation value of its
position, equal to the particle’s actual speed), gives a value for the kinetic energy KE of
such a particle which has non-zero rest mass and is moving with a speed small
compared to that of light which is about 2 times larger than its actual kinetic energy, so
setting hf = the actual kinetic energy gives a value for hf which is about 2 times too
small. Indeed, the Relativistic (true) kinetic energy rKE of a particle with rest mass m0
is given by rKE = total energy (excluding potential energy) - rest mass equivalent energy =
m0c2 [1/√(1-s2/c2) - 1] = mc2 -m0c2, where m is the particle’s Relativistic (moving) mass;
then the limit of rKE/ms2 as s goes to c is 1, while as s goes to 0 it is ½.
Why hasn’t this obvious theoretical error been seen by others? Maybe the fact that it is
an error is known to most competent physicists, but for some reason the error nevertheless
is made in almost all introductory discussions on the Internet of solutions to
Schrodinger’s equation, and, I imagine, since it is omnipresent in such discussions, it is
still being taught to beginning q.m. students in universities. On the other hand, maybe I
am still making the error that years ago I thought I was making, whatever that is.
quantum mechanics, but I assumed it was due to an error on my part which I, however,
couldn’t locate, and I didn’t ask my course instructor about it. Recently, when
researching a q.m. question on the Internet, I rediscovered it: Both in my textbooks years
ago and on the Internet now, (non-normalizable) plane-wave solutions to the
non-Relativistic, 1-dimensional, but easily extendable to 3 dimensions, Schrodinger
wave equation for any free particle PRTCL with non-zero rest mass (which always
moves at less than the speed of light), such as an electron, are given as proportional to
Ψ = exp(i[kx-ωt]), where k = 2π/λ, λ being Ψ’s wavelength, ω = 2πf, where f is Ψ’s
frequency, i is the imaginary constant, x is the length coordinate, and t is time. The
problem arises from the identification of h/λ with the classical momentum p = ms of
PRTCL, and hf with its classical kinetic energy KE = p2/2m = ms2/2, where m is PRTCL’s
mass, s is its speed, and h is Planck’s Constant. Then Ψ = exp(2πi[px -KEt]/h) = exp(2π
i[msx - ms2 t/2]/h), and if t increases one unit, the term it is in increases by ms2 /2, so for
Ψ to have the same phase as before, x must increase by ms2/2ms = s/2 units, so the
phase velocity (speed) of Ψ is s/2. This seems to be at variance with the speed of
PRTCL's being s, but it might be objected that Ψ is not a physically realistic wavefunction,
being non-normalizable. However, Ψ can be modified by smoothly limiting it to be zero
outside of some large but finite region R, but to be the same as before inside of some
region R’ inside but nearly as big as R, so that the modified wave function Ψ' is
normalizable, and so that Ψ' is the wave function of some free particle mPRTCL with
only slightly uncertain momentum and speed, with momentum expectation value p and
speed expectation value s, while the phase velocity of the main Fourier component of Ψ′
and the speed of change of its position expectation value is s/2.
The cause of this discrepancy seems to be that KE = hf, which Louis de Broglie, the
inventor of the idea of the wave function of a particle, is supposed to have assumed to
be true for all particles, is true for photons, which move with the speed of light, and all of
whose energy E = mc2 is kinetic energy, none being their rest mass equivalent energy,
so their KE = mc2 = ms2 (E = hf for a photon was Planck’s quantum hypothesis, which began
quantum mechanics, made to avoid the Ultraviolet Catastrophe of then-current
theory [maybe the UC wasn’t implied by then-current-theory, and I’ll post a thread about
that]), but KE = hf, where f is the true frequency of the wave function of a particle in a
momentum eigenstate (i.e., the value of f in the equation above which is necessary to
make the phase velocity, or the speed of the movement of the expectation value of its
position, equal to the particle’s actual speed), gives a value for the kinetic energy KE of
such a particle which has non-zero rest mass and is moving with a speed small
compared to that of light which is about 2 times larger than its actual kinetic energy, so
setting hf = the actual kinetic energy gives a value for hf which is about 2 times too
small. Indeed, the Relativistic (true) kinetic energy rKE of a particle with rest mass m0
is given by rKE = total energy (excluding potential energy) - rest mass equivalent energy =
m0c2 [1/√(1-s2/c2) - 1] = mc2 -m0c2, where m is the particle’s Relativistic (moving) mass;
then the limit of rKE/ms2 as s goes to c is 1, while as s goes to 0 it is ½.
Why hasn’t this obvious theoretical error been seen by others? Maybe the fact that it is
an error is known to most competent physicists, but for some reason the error nevertheless
is made in almost all introductory discussions on the Internet of solutions to
Schrodinger’s equation, and, I imagine, since it is omnipresent in such discussions, it is
still being taught to beginning q.m. students in universities. On the other hand, maybe I
am still making the error that years ago I thought I was making, whatever that is.