Review of Early antum Mechanics Formalization of antum Mechanics
Physics 280 antum Mechanics Lecture Spring 2015
Dr. Jones1
1Department of Physics Drexel University
August 3, 2016
Dr. Jones Physics 280 antum Mechanics Lecture Schrodinger’s¨ Wave Equation Heisenberg Uncertainty Strange Consquences Strange Contradictions
Review of Early antum Mechanics Formalization of antum Mechanics Objectives
Review Early antum Mechanics
Dr. Jones Physics 280 antum Mechanics Lecture Heisenberg Uncertainty Strange Consquences Strange Contradictions
Review of Early antum Mechanics Formalization of antum Mechanics Objectives
Review Early antum Mechanics Schrodinger’s¨ Wave Equation
Dr. Jones Physics 280 antum Mechanics Lecture Strange Consquences Strange Contradictions
Review of Early antum Mechanics Formalization of antum Mechanics Objectives
Review Early antum Mechanics Schrodinger’s¨ Wave Equation Heisenberg Uncertainty
Dr. Jones Physics 280 antum Mechanics Lecture Strange Contradictions
Review of Early antum Mechanics Formalization of antum Mechanics Objectives
Review Early antum Mechanics Schrodinger’s¨ Wave Equation Heisenberg Uncertainty Strange Consquences
Dr. Jones Physics 280 antum Mechanics Lecture Review of Early antum Mechanics Formalization of antum Mechanics Objectives
Review Early antum Mechanics Schrodinger’s¨ Wave Equation Heisenberg Uncertainty Strange Consquences Strange Contradictions
Dr. Jones Physics 280 antum Mechanics Lecture Classical Physics predicts that the intensity of radiation from blackbody-like sources goes to infinity with decreasing wavelength, which is experimentally shown to be untrue and is impossible anyway. Planck uses a mathematical trick for the first time ever–energy quantization: E = hf to ”fix” the equations. Einstein takes this result at face value and shows that photons are quantized in the same way, explaining the photoelectric eect. Bohr adapts the idea of quantization to ”explain” the Rydberg equation and why the emission/absorption spectrum is quantized. He quantizes angular momentum.
Review of Early antum Mechanics Formalization of antum Mechanics So far..
Electrical charges are quantized in the form of electrons and protons, nothing is smaller than ±e.
Dr. Jones Physics 280 antum Mechanics Lecture Planck uses a mathematical trick for the first time ever–energy quantization: E = hf to ”fix” the equations. Einstein takes this result at face value and shows that photons are quantized in the same way, explaining the photoelectric eect. Bohr adapts the idea of quantization to ”explain” the Rydberg equation and why the emission/absorption spectrum is quantized. He quantizes angular momentum.
Review of Early antum Mechanics Formalization of antum Mechanics So far..
Electrical charges are quantized in the form of electrons and protons, nothing is smaller than ±e. Classical Physics predicts that the intensity of radiation from blackbody-like sources goes to infinity with decreasing wavelength, which is experimentally shown to be untrue and is impossible anyway.
Dr. Jones Physics 280 antum Mechanics Lecture Einstein takes this result at face value and shows that photons are quantized in the same way, explaining the photoelectric eect. Bohr adapts the idea of quantization to ”explain” the Rydberg equation and why the emission/absorption spectrum is quantized. He quantizes angular momentum.
Review of Early antum Mechanics Formalization of antum Mechanics So far..
Electrical charges are quantized in the form of electrons and protons, nothing is smaller than ±e. Classical Physics predicts that the intensity of radiation from blackbody-like sources goes to infinity with decreasing wavelength, which is experimentally shown to be untrue and is impossible anyway. Planck uses a mathematical trick for the first time ever–energy quantization: E = hf to ”fix” the equations.
Dr. Jones Physics 280 antum Mechanics Lecture Bohr adapts the idea of quantization to ”explain” the Rydberg equation and why the emission/absorption spectrum is quantized. He quantizes angular momentum.
Review of Early antum Mechanics Formalization of antum Mechanics So far..
Electrical charges are quantized in the form of electrons and protons, nothing is smaller than ±e. Classical Physics predicts that the intensity of radiation from blackbody-like sources goes to infinity with decreasing wavelength, which is experimentally shown to be untrue and is impossible anyway. Planck uses a mathematical trick for the first time ever–energy quantization: E = hf to ”fix” the equations. Einstein takes this result at face value and shows that photons are quantized in the same way, explaining the photoelectric eect.
Dr. Jones Physics 280 antum Mechanics Lecture Review of Early antum Mechanics Formalization of antum Mechanics So far..
Electrical charges are quantized in the form of electrons and protons, nothing is smaller than ±e. Classical Physics predicts that the intensity of radiation from blackbody-like sources goes to infinity with decreasing wavelength, which is experimentally shown to be untrue and is impossible anyway. Planck uses a mathematical trick for the first time ever–energy quantization: E = hf to ”fix” the equations. Einstein takes this result at face value and shows that photons are quantized in the same way, explaining the photoelectric eect. Bohr adapts the idea of quantization to ”explain” the Rydberg equation and why the emission/absorption spectrum is quantized. He quantizes angular momentum.
Dr. Jones Physics 280 antum Mechanics Lecture Review of Early antum Mechanics Formalization of antum Mechanics
A new truth Energy is quantized.
Dr. Jones Physics 280 antum Mechanics Lecture We use blackbody radiation as a proxy for the more complex thermal radiation. Wien’s law can explain the shi of the peak lewards with temperature.
−2 λmax T = 0.2898 × 10 mK
Review of Early antum Mechanics Formalization of antum Mechanics The Ultraviolet Catastrophe
Thermal radiation of a body depends upon the temperature and wavelength.
Dr. Jones Physics 280 antum Mechanics Lecture Wien’s law can explain the shi of the peak lewards with temperature.
−2 λmax T = 0.2898 × 10 mK
Review of Early antum Mechanics Formalization of antum Mechanics The Ultraviolet Catastrophe
Thermal radiation of a body depends upon the temperature and wavelength. We use blackbody radiation as a proxy for the more complex thermal radiation.
Dr. Jones Physics 280 antum Mechanics Lecture Review of Early antum Mechanics Formalization of antum Mechanics The Ultraviolet Catastrophe
Thermal radiation of a body depends upon the temperature and wavelength. We use blackbody radiation as a proxy for the more complex thermal radiation. Wien’s law can explain the shi of the peak lewards with temperature.
−2 λmax T = 0.2898 × 10 mK Dr. Jones Physics 280 antum Mechanics Lecture Review of Early antum Mechanics Formalization of antum Mechanics Why we can approximate some glowing bodies as blackbody radiation
Dr. Jones Physics 280 antum Mechanics Lecture Combining these laws yields the Rayleigh-Jeans prediction:
2πck T I(λ, T) = B λ4 Classical physics could take us no further, and it was wrong.
Review of Early antum Mechanics Formalization of antum Mechanics The Ultraviolet Catastrophe
Stefan’s law predicts the power emied by blackbody radiation:
P = σAeT 4
Dr. Jones Physics 280 antum Mechanics Lecture Classical physics could take us no further, and it was wrong.
Review of Early antum Mechanics Formalization of antum Mechanics The Ultraviolet Catastrophe
Stefan’s law predicts the power emied by blackbody radiation:
P = σAeT 4
Combining these laws yields the Rayleigh-Jeans prediction:
2πck T I(λ, T) = B λ4
Dr. Jones Physics 280 antum Mechanics Lecture Review of Early antum Mechanics Formalization of antum Mechanics The Ultraviolet Catastrophe
Stefan’s law predicts the power emied by blackbody radiation:
P = σAeT 4
Combining these laws yields the Rayleigh-Jeans prediction:
2πck T I(λ, T) = B λ4 Classical physics could take us no further, and it was wrong.
Dr. Jones Physics 280 antum Mechanics Lecture n is a positive integer, the quantum number, f is frequency of vibration, h is a new constant he fit to experimental results:
h = 6.626 × 10−34J · s
2πhc2 I(λ, T) = hc λ5 e λkBT − 1
Review of Early antum Mechanics Formalization of antum Mechanics The Ultraviolet Catastrophe: Resolved Reluctantly
Planck’s leap: Supposing that blackbody radiation was emied by ’resonators’, these resonators could only have energy in discreet quantity: En = nhf
Dr. Jones Physics 280 antum Mechanics Lecture 2πhc2 I(λ, T) = hc λ5 e λkBT − 1
Review of Early antum Mechanics Formalization of antum Mechanics The Ultraviolet Catastrophe: Resolved Reluctantly
Planck’s leap: Supposing that blackbody radiation was emied by ’resonators’, these resonators could only have energy in discreet quantity: En = nhf n is a positive integer, the quantum number, f is frequency of vibration, h is a new constant he fit to experimental results:
h = 6.626 × 10−34J · s
Dr. Jones Physics 280 antum Mechanics Lecture Review of Early antum Mechanics Formalization of antum Mechanics The Ultraviolet Catastrophe: Resolved Reluctantly
Planck’s leap: Supposing that blackbody radiation was emied by ’resonators’, these resonators could only have energy in discreet quantity: En = nhf n is a positive integer, the quantum number, f is frequency of vibration, h is a new constant he fit to experimental results:
h = 6.626 × 10−34J · s
2πhc2 I(λ, T) = hc λ5 e λkBT − 1
Dr. Jones Physics 280 antum Mechanics Lecture Another mystery dominated atomic physics–nobody could explain the spectra of gasses. We might expect a continuous distribution of wavelengths, but instead we find discrete line spectrum called the emission spectrum. Passing white light through gasses result in discrete missing wave- lengths, this is called the absorption spectrum.
Review of Early antum Mechanics Formalization of antum Mechanics Atomic mystery
antization of energy solved the Blackbody Radiation problem and the Photoelectric Eect. It solved the scaering problem (Compton eect) and although nobody could quite make sense of it,
Dr. Jones Physics 280 antum Mechanics Lecture We might expect a continuous distribution of wavelengths, but instead we find discrete line spectrum called the emission spectrum. Passing white light through gasses result in discrete missing wave- lengths, this is called the absorption spectrum.
Review of Early antum Mechanics Formalization of antum Mechanics Atomic mystery
antization of energy solved the Blackbody Radiation problem and the Photoelectric Eect. It solved the scaering problem (Compton eect) and although nobody could quite make sense of it, Another mystery dominated atomic physics–nobody could explain the spectra of gasses.
Dr. Jones Physics 280 antum Mechanics Lecture Passing white light through gasses result in discrete missing wave- lengths, this is called the absorption spectrum.
Review of Early antum Mechanics Formalization of antum Mechanics Atomic mystery
antization of energy solved the Blackbody Radiation problem and the Photoelectric Eect. It solved the scaering problem (Compton eect) and although nobody could quite make sense of it, Another mystery dominated atomic physics–nobody could explain the spectra of gasses. We might expect a continuous distribution of wavelengths, but instead we find discrete line spectrum called the emission spectrum.
Dr. Jones Physics 280 antum Mechanics Lecture Review of Early antum Mechanics Formalization of antum Mechanics Atomic mystery
antization of energy solved the Blackbody Radiation problem and the Photoelectric Eect. It solved the scaering problem (Compton eect) and although nobody could quite make sense of it, Another mystery dominated atomic physics–nobody could explain the spectra of gasses. We might expect a continuous distribution of wavelengths, but instead we find discrete line spectrum called the emission spectrum. Passing white light through gasses result in discrete missing wave- lengths, this is called the absorption spectrum.
Dr. Jones Physics 280 antum Mechanics Lecture 19th century physicists can’t explain these spectra.
Review of Early antum Mechanics Formalization of antum Mechanics Atomic mystery
Example emission (hydrogen, mercury, and neon) and absorption spectrum for hydrogen:
Dr. Jones Physics 280 antum Mechanics Lecture Review of Early antum Mechanics Formalization of antum Mechanics Atomic mystery
Example emission (hydrogen, mercury, and neon) and absorption spectrum for hydrogen:
19th century physicists can’t explain these spectra.
Dr. Jones Physics 280 antum Mechanics Lecture RH is a constant called the Rydberg constant and is equal to 1.0973732 ×107m−1. The shortest wavelength is found when n → ∞ is called the series limit with wavelength 364.6 nm (ultraviolet).
Review of Early antum Mechanics Formalization of antum Mechanics Atomic mystery
Balmer found an empirical equation that correctly predicted the wavelengths of four of the visible emission lines; Rydberg expanded this equation to find all emission lines:
1 1 1 = R − , n = 3, 4, 5, ··· λ H 22 n2
Dr. Jones Physics 280 antum Mechanics Lecture The shortest wavelength is found when n → ∞ is called the series limit with wavelength 364.6 nm (ultraviolet).
Review of Early antum Mechanics Formalization of antum Mechanics Atomic mystery
Balmer found an empirical equation that correctly predicted the wavelengths of four of the visible emission lines; Rydberg expanded this equation to find all emission lines:
1 1 1 = R − , n = 3, 4, 5, ··· λ H 22 n2
RH is a constant called the Rydberg constant and is equal to 1.0973732 ×107m−1.
Dr. Jones Physics 280 antum Mechanics Lecture Review of Early antum Mechanics Formalization of antum Mechanics Atomic mystery
Balmer found an empirical equation that correctly predicted the wavelengths of four of the visible emission lines; Rydberg expanded this equation to find all emission lines:
1 1 1 = R − , n = 3, 4, 5, ··· λ H 22 n2
RH is a constant called the Rydberg constant and is equal to 1.0973732 ×107m−1. The shortest wavelength is found when n → ∞ is called the series limit with wavelength 364.6 nm (ultraviolet).
Dr. Jones Physics 280 antum Mechanics Lecture Review of Early antum Mechanics Formalization of antum Mechanics Atomic mystery
Other physicists started experimenting with these numbers and found similar equations that described other lines in the spectrum:
1 1 Lyman series: = R 1 − , n = 2, 3, 4, ··· λ H n2 1 1 1 Paschen series: = R − , n = 4, 5, 6, ··· λ H 32 n2 1 1 1 Bracket series: = R − , n = 5, 6, 7, ··· λ H 42 n2
It all works very well, but nobody knows why.
Dr. Jones Physics 280 antum Mechanics Lecture Review of Early antum Mechanics Formalization of antum Mechanics Atomic mystery
Dr. Jones Physics 280 antum Mechanics Lecture Review of Early antum Mechanics Formalization of antum Mechanics antum solution
A physicists named Niels Bohr saw that energy quantization had solved the blackbody and photoelectric problem, and he wondered if it could solve the mystery of the atomic spectrum lines as well. Here is a basic outline of his reasoning.
Dr. Jones Physics 280 antum Mechanics Lecture So far, so classical.
Review of Early antum Mechanics Formalization of antum Mechanics antum solution
Assume the classical model of the electron orbiting the nucleus of the hydrogen atom under electrical forces. In this case the total energy is 1 e2 E = K + U = m v2 − k 2 e e r and since we assume the electron is going in a circle, the electric force must act as a centripetal force: k e2 m v2 k e2 e = e =⇒ 2 = e 2 v r r mer Using this value for velocity, we have kinetic energy: 1 k e2 k e2 K = m v2 = e and E = − e 2 e 2r 2r
Dr. Jones Physics 280 antum Mechanics Lecture Review of Early antum Mechanics Formalization of antum Mechanics antum solution
Assume the classical model of the electron orbiting the nucleus of the hydrogen atom under electrical forces. In this case the total energy is 1 e2 E = K + U = m v2 − k 2 e e r and since we assume the electron is going in a circle, the electric force must act as a centripetal force: k e2 m v2 k e2 e = e =⇒ 2 = e 2 v r r mer Using this value for velocity, we have kinetic energy: 1 k e2 k e2 K = m v2 = e and E = − e 2 e 2r 2r So far, so classical.
Dr. Jones Physics 280 antum Mechanics Lecture Review of Early antum Mechanics Formalization of antum Mechanics antum solution
Next comes the quantum step. Assume that only certain orbits are stable (called stationary states, which validates are previous assumption of using classical physics). The atom emits radiation when it makes a quantum leap from one state to another. The change in its energy aer making this leap is quantized:
Ef −Ei = hf , positive value means absorbed, negative value means emied
Dr. Jones Physics 280 antum Mechanics Lecture Now lets apply this radical concept:
n2 2 k e2 n2 2 2 = ~ = e =⇒ = ~ , = , , v 2 2 rn 2 n 1 2 3 mer mer mekee
Review of Early antum Mechanics Formalization of antum Mechanics antum solution
Bohr next a new leap: quantizing the electron’s orbital angular momentum: h mevr = n = n~ 2π where ~ = h/2π. The energy quantization was enough to fluster some, but this laer assumption was a radically new expansion of the concept of quantization.
Dr. Jones Physics 280 antum Mechanics Lecture Review of Early antum Mechanics Formalization of antum Mechanics antum solution
Bohr next a new leap: quantizing the electron’s orbital angular momentum: h mevr = n = n~ 2π where ~ = h/2π. The energy quantization was enough to fluster some, but this laer assumption was a radically new expansion of the concept of quantization. Now lets apply this radical concept:
n2 2 k e2 n2 2 2 = ~ = e =⇒ = ~ , = , , v 2 2 rn 2 n 1 2 3 mer mer mekee
Dr. Jones Physics 280 antum Mechanics Lecture Review of Early antum Mechanics Formalization of antum Mechanics antum solution
Call the orbit with the smallest radius the Bohr radius (when n = 1): 2 = ~ = . a0 2 0 0529nm mekee Plug this all back into the energy equation to find: k e2 1 = − e , = , , , ··· En 2 n 1 2 3 2ao n or numerically: 13.606eV E = − , n = 1, 2, 3, ··· n n2 Now let’s find out the frequency of light emied from a quantum leap: 2 ! Ei − Ef kee 1 1 f = = 2 − 2 h 2a0h nf ni Dr. Jones Physics 280 antum Mechanics Lecture antum mechanics had theoretically predicted the value of RH. At this point, aempts to rescue classical physics were beginning to look hopeless. We see it again: integers describing physical reality–one of the key signatures of quantum mechanics. This made most physicists uncomfortable, but also energized by the revolutionary spirit in the air.
Review of Early antum Mechanics Formalization of antum Mechanics antum solution
That looks familiar… 2 ! 1 f kee 1 1 = = 2 − 2 λ c 2a0hc nf ni Plugging in all of our numbers: 2 kee 7 −1 = 1.0973732 × 10 m = RH 2a0hc
Dr. Jones Physics 280 antum Mechanics Lecture We see it again: integers describing physical reality–one of the key signatures of quantum mechanics. This made most physicists uncomfortable, but also energized by the revolutionary spirit in the air.
Review of Early antum Mechanics Formalization of antum Mechanics antum solution
That looks familiar… 2 ! 1 f kee 1 1 = = 2 − 2 λ c 2a0hc nf ni Plugging in all of our numbers: 2 kee 7 −1 = 1.0973732 × 10 m = RH 2a0hc antum mechanics had theoretically predicted the value of RH. At this point, aempts to rescue classical physics were beginning to look hopeless.
Dr. Jones Physics 280 antum Mechanics Lecture Review of Early antum Mechanics Formalization of antum Mechanics antum solution
That looks familiar… 2 ! 1 f kee 1 1 = = 2 − 2 λ c 2a0hc nf ni Plugging in all of our numbers: 2 kee 7 −1 = 1.0973732 × 10 m = RH 2a0hc antum mechanics had theoretically predicted the value of RH. At this point, aempts to rescue classical physics were beginning to look hopeless. We see it again: integers describing physical reality–one of the key signatures of quantum mechanics. This made most physicists uncomfortable, but also energized by the revolutionary spirit in the air.
Dr. Jones Physics 280 antum Mechanics Lecture Einstein proposes that light is quantized (1905), solved photo-electric eect Bohr proposes electron orbital angular momentum is quantized −13.606eV (1913), solves hydrogen spectrum problem, En = n2 de Broglie (1923/24) proposes that if photons are waves and particles, so are massive particles like electrons:
E hf h h p = mv = = = → λ = c c λ mv
Review of Early antum Mechanics Formalization of antum Mechanics Summary:
Planck predicts quantization (1900): E = nhf , solves ultraviolet catastrophe
Dr. Jones Physics 280 antum Mechanics Lecture Bohr proposes electron orbital angular momentum is quantized −13.606eV (1913), solves hydrogen spectrum problem, En = n2 de Broglie (1923/24) proposes that if photons are waves and particles, so are massive particles like electrons:
E hf h h p = mv = = = → λ = c c λ mv
Review of Early antum Mechanics Formalization of antum Mechanics Summary:
Planck predicts quantization (1900): E = nhf , solves ultraviolet catastrophe Einstein proposes that light is quantized (1905), solved photo-electric eect
Dr. Jones Physics 280 antum Mechanics Lecture de Broglie (1923/24) proposes that if photons are waves and particles, so are massive particles like electrons:
E hf h h p = mv = = = → λ = c c λ mv
Review of Early antum Mechanics Formalization of antum Mechanics Summary:
Planck predicts quantization (1900): E = nhf , solves ultraviolet catastrophe Einstein proposes that light is quantized (1905), solved photo-electric eect Bohr proposes electron orbital angular momentum is quantized −13.606eV (1913), solves hydrogen spectrum problem, En = n2
Dr. Jones Physics 280 antum Mechanics Lecture Review of Early antum Mechanics Formalization of antum Mechanics Summary:
Planck predicts quantization (1900): E = nhf , solves ultraviolet catastrophe Einstein proposes that light is quantized (1905), solved photo-electric eect Bohr proposes electron orbital angular momentum is quantized −13.606eV (1913), solves hydrogen spectrum problem, En = n2 de Broglie (1923/24) proposes that if photons are waves and particles, so are massive particles like electrons:
E hf h h p = mv = = = → λ = c c λ mv
Dr. Jones Physics 280 antum Mechanics Lecture For light, the intensity of the wave is proportional to the square of the electric field. The more intense the light, the more photons we can find, and so the number of photons is proportional to the intensity is proportional to the square of the electric field (the EM wave amplitude). Put another way, the square of the electric field amplitude is proportional to the probability that we will find a photon at a certain region. We seek a maer wave such that Ψ2 is proportional the probability of finding an electron at a certain place and time.
Review of Early antum Mechanics Formalization of antum Mechanics Schrodinger’s¨ formalization
If particles are waves, how do we discuss their amplitudes and displacements like we do for classical waves? We need some sort of wave function to do so, which we write as Ψ.
Dr. Jones Physics 280 antum Mechanics Lecture Put another way, the square of the electric field amplitude is proportional to the probability that we will find a photon at a certain region. We seek a maer wave such that Ψ2 is proportional the probability of finding an electron at a certain place and time.
Review of Early antum Mechanics Formalization of antum Mechanics Schrodinger’s¨ formalization
If particles are waves, how do we discuss their amplitudes and displacements like we do for classical waves? We need some sort of wave function to do so, which we write as Ψ. For light, the intensity of the wave is proportional to the square of the electric field. The more intense the light, the more photons we can find, and so the number of photons is proportional to the intensity is proportional to the square of the electric field (the EM wave amplitude).
Dr. Jones Physics 280 antum Mechanics Lecture We seek a maer wave such that Ψ2 is proportional the probability of finding an electron at a certain place and time.
Review of Early antum Mechanics Formalization of antum Mechanics Schrodinger’s¨ formalization
If particles are waves, how do we discuss their amplitudes and displacements like we do for classical waves? We need some sort of wave function to do so, which we write as Ψ. For light, the intensity of the wave is proportional to the square of the electric field. The more intense the light, the more photons we can find, and so the number of photons is proportional to the intensity is proportional to the square of the electric field (the EM wave amplitude). Put another way, the square of the electric field amplitude is proportional to the probability that we will find a photon at a certain region.
Dr. Jones Physics 280 antum Mechanics Lecture Review of Early antum Mechanics Formalization of antum Mechanics Schrodinger’s¨ formalization
If particles are waves, how do we discuss their amplitudes and displacements like we do for classical waves? We need some sort of wave function to do so, which we write as Ψ. For light, the intensity of the wave is proportional to the square of the electric field. The more intense the light, the more photons we can find, and so the number of photons is proportional to the intensity is proportional to the square of the electric field (the EM wave amplitude). Put another way, the square of the electric field amplitude is proportional to the probability that we will find a photon at a certain region. We seek a maer wave such that Ψ2 is proportional the probability of finding an electron at a certain place and time.
Dr. Jones Physics 280 antum Mechanics Lecture This is called the Schrodinger¨ equation. For most particles, we can separate the time-dependence so that Ψ(r, t) = ψ(r)e−iωt This is a probabilistic equation, where the probability of finding a particle whose wave is ψ within any given region of space is P(x, y, z)dV =| ψ |2 dV. What is maer then? A particle? A wave? Is ψ the true manifestation of maer? ψ is the true manifestation of maer.
Review of Early antum Mechanics Formalization of antum Mechanics Schrodinger’s¨ formalization
Schrodinger¨ found the equation that properly describes how these maer waves behave: 2 d2ψ − ~ + Uψ = Eψ 2m dx2
Dr. Jones Physics 280 antum Mechanics Lecture For most particles, we can separate the time-dependence so that Ψ(r, t) = ψ(r)e−iωt This is a probabilistic equation, where the probability of finding a particle whose wave is ψ within any given region of space is P(x, y, z)dV =| ψ |2 dV. What is maer then? A particle? A wave? Is ψ the true manifestation of maer? ψ is the true manifestation of maer.
Review of Early antum Mechanics Formalization of antum Mechanics Schrodinger’s¨ formalization
Schrodinger¨ found the equation that properly describes how these maer waves behave: 2 d2ψ − ~ + Uψ = Eψ 2m dx2 This is called the Schrodinger¨ equation.
Dr. Jones Physics 280 antum Mechanics Lecture This is a probabilistic equation, where the probability of finding a particle whose wave is ψ within any given region of space is P(x, y, z)dV =| ψ |2 dV. What is maer then? A particle? A wave? Is ψ the true manifestation of maer? ψ is the true manifestation of maer.
Review of Early antum Mechanics Formalization of antum Mechanics Schrodinger’s¨ formalization
Schrodinger¨ found the equation that properly describes how these maer waves behave: 2 d2ψ − ~ + Uψ = Eψ 2m dx2 This is called the Schrodinger¨ equation. For most particles, we can separate the time-dependence so that Ψ(r, t) = ψ(r)e−iωt
Dr. Jones Physics 280 antum Mechanics Lecture What is maer then? A particle? A wave? Is ψ the true manifestation of maer? ψ is the true manifestation of maer.
Review of Early antum Mechanics Formalization of antum Mechanics Schrodinger’s¨ formalization
Schrodinger¨ found the equation that properly describes how these maer waves behave: 2 d2ψ − ~ + Uψ = Eψ 2m dx2 This is called the Schrodinger¨ equation. For most particles, we can separate the time-dependence so that Ψ(r, t) = ψ(r)e−iωt This is a probabilistic equation, where the probability of finding a particle whose wave is ψ within any given region of space is P(x, y, z)dV =| ψ |2 dV.
Dr. Jones Physics 280 antum Mechanics Lecture ψ is the true manifestation of maer.
Review of Early antum Mechanics Formalization of antum Mechanics Schrodinger’s¨ formalization
Schrodinger¨ found the equation that properly describes how these maer waves behave: 2 d2ψ − ~ + Uψ = Eψ 2m dx2 This is called the Schrodinger¨ equation. For most particles, we can separate the time-dependence so that Ψ(r, t) = ψ(r)e−iωt This is a probabilistic equation, where the probability of finding a particle whose wave is ψ within any given region of space is P(x, y, z)dV =| ψ |2 dV. What is maer then? A particle? A wave? Is ψ the true manifestation of maer?
Dr. Jones Physics 280 antum Mechanics Lecture Review of Early antum Mechanics Formalization of antum Mechanics Schrodinger’s¨ formalization
Schrodinger¨ found the equation that properly describes how these maer waves behave: 2 d2ψ − ~ + Uψ = Eψ 2m dx2 This is called the Schrodinger¨ equation. For most particles, we can separate the time-dependence so that Ψ(r, t) = ψ(r)e−iωt This is a probabilistic equation, where the probability of finding a particle whose wave is ψ within any given region of space is P(x, y, z)dV =| ψ |2 dV. What is maer then? A particle? A wave? Is ψ the true manifestation of maer? ψ is the true manifestation of maer.
Dr. Jones Physics 280 antum Mechanics Lecture It can be show that the ”standard deviation in the results of repeated measurements on identically prepared systems”, or the ”uncertainty”, has the following relation:
∆x∆p ≥ ~ 2 This is the Heisenberg Uncertainty Principal.
Review of Early antum Mechanics Formalization of antum Mechanics Probability
For any probability function, the integral of the entire function over infinity must be equal to one:
Z +∞ |ψ(x)|2 dx = 1 −∞
Dr. Jones Physics 280 antum Mechanics Lecture This is the Heisenberg Uncertainty Principal.
Review of Early antum Mechanics Formalization of antum Mechanics Probability
For any probability function, the integral of the entire function over infinity must be equal to one:
Z +∞ |ψ(x)|2 dx = 1 −∞ It can be show that the ”standard deviation in the results of repeated measurements on identically prepared systems”, or the ”uncertainty”, has the following relation:
∆x∆p ≥ ~ 2
Dr. Jones Physics 280 antum Mechanics Lecture Review of Early antum Mechanics Formalization of antum Mechanics Probability
For any probability function, the integral of the entire function over infinity must be equal to one:
Z +∞ |ψ(x)|2 dx = 1 −∞ It can be show that the ”standard deviation in the results of repeated measurements on identically prepared systems”, or the ”uncertainty”, has the following relation:
∆x∆p ≥ ~ 2 This is the Heisenberg Uncertainty Principal.
Dr. Jones Physics 280 antum Mechanics Lecture Review of Early antum Mechanics Formalization of antum Mechanics Probability
When there is a barrier which has too much energy for a particle to pass through in terms of classical physics, quantum mechanics correctly predicts that there is still a probability it will pass through. This is called tunneling.
Dr. Jones Physics 280 antum Mechanics Lecture Review of Early antum Mechanics Formalization of antum Mechanics
Dr. Jones Physics 280 antum Mechanics Lecture Review of Early antum Mechanics Formalization of antum Mechanics
Dr. Jones Physics 280 antum Mechanics Lecture Review of Early antum Mechanics Formalization of antum Mechanics
Dr. Jones Physics 280 antum Mechanics Lecture