• The free particle wavefunction
• Properties of the free particle
• Wavepackets, real free particles
• Phase & group velocities
Free particles
Carlo Segre (Illinois Tech) PHYS 405 - Fundamentals of Quantum Theory I The free particle • Properties of the free particle
• Wavepackets, real free particles
• Phase & group velocities
Free particles
• The free particle wavefunction
Carlo Segre (Illinois Tech) PHYS 405 - Fundamentals of Quantum Theory I The free particle • Wavepackets, real free particles
• Phase & group velocities
Free particles
• The free particle wavefunction
• Properties of the free particle
Carlo Segre (Illinois Tech) PHYS 405 - Fundamentals of Quantum Theory I The free particle • Phase & group velocities
Free particles
• The free particle wavefunction
• Properties of the free particle
• Wavepackets, real free particles
Carlo Segre (Illinois Tech) PHYS 405 - Fundamentals of Quantum Theory I The free particle Free particles
• The free particle wavefunction
• Properties of the free particle
• Wavepackets, real free particles
• Phase & group velocities
Carlo Segre (Illinois Tech) PHYS 405 - Fundamentals of Quantum Theory I The free particle √ 2mE ≡ Aei(kx−ωt), k = ± ~
2 d2ψ − ~ = Eψ 2m dx2 d2ψ 2m = − Eψ = −k2ψ dx2 ~2 The free particle Hamiltonian has V ≡ 0 everywhere In this case, however, our trial solution needs to be of the form ψ(x) = Aeikx + Be−ikx
Adding the time-dependent portion e−i(E/~)t = e−iωt
Ψ(x, t) = Aei(kx−ωt) + Be−i(kx+ωt)
These are simply traveling waves in ±x directions so we can simplify
The free particle wavefunction
The free particle would seem to be the sim- plest case, however, in quantum mechanics it is actually rather complex and needs to be treated carefully.
Carlo Segre (Illinois Tech) PHYS 405 - Fundamentals of Quantum Theory I The free particle √ 2mE ≡ Aei(kx−ωt), k = ± ~
2 d2ψ − ~ = Eψ 2m dx2 d2ψ 2m = − Eψ = −k2ψ dx2 ~2
In this case, however, our trial solution needs to be of the form ψ(x) = Aeikx + Be−ikx
Adding the time-dependent portion e−i(E/~)t = e−iωt
Ψ(x, t) = Aei(kx−ωt) + Be−i(kx+ωt)
These are simply traveling waves in ±x directions so we can simplify
The free particle wavefunction
The free particle would seem to be the sim- plest case, however, in quantum mechanics it is actually rather complex and needs to be treated carefully. The free particle Hamiltonian has V ≡ 0 everywhere
Carlo Segre (Illinois Tech) PHYS 405 - Fundamentals of Quantum Theory I The free particle √ 2mE ≡ Aei(kx−ωt), k = ± ~
d2ψ 2m = − Eψ = −k2ψ dx2 ~2
In this case, however, our trial solution needs to be of the form ψ(x) = Aeikx + Be−ikx
Adding the time-dependent portion e−i(E/~)t = e−iωt
Ψ(x, t) = Aei(kx−ωt) + Be−i(kx+ωt)
These are simply traveling waves in ±x directions so we can simplify
The free particle wavefunction
2 d2ψ The free particle would seem to be the sim- − ~ = Eψ plest case, however, in quantum mechanics 2m dx2 it is actually rather complex and needs to be treated carefully. The free particle Hamiltonian has V ≡ 0 everywhere
Carlo Segre (Illinois Tech) PHYS 405 - Fundamentals of Quantum Theory I The free particle √ 2mE ≡ Aei(kx−ωt), k = ± ~
= −k2ψ
In this case, however, our trial solution needs to be of the form ψ(x) = Aeikx + Be−ikx
Adding the time-dependent portion e−i(E/~)t = e−iωt
Ψ(x, t) = Aei(kx−ωt) + Be−i(kx+ωt)
These are simply traveling waves in ±x directions so we can simplify
The free particle wavefunction
2 d2ψ The free particle would seem to be the sim- − ~ = Eψ plest case, however, in quantum mechanics 2m dx2 it is actually rather complex and needs to be d2ψ 2m = − Eψ treated carefully. dx2 ~2 The free particle Hamiltonian has V ≡ 0 everywhere
Carlo Segre (Illinois Tech) PHYS 405 - Fundamentals of Quantum Theory I The free particle √ 2mE ≡ Aei(kx−ωt), k = ± ~
In this case, however, our trial solution needs to be of the form ψ(x) = Aeikx + Be−ikx
Adding the time-dependent portion e−i(E/~)t = e−iωt
Ψ(x, t) = Aei(kx−ωt) + Be−i(kx+ωt)
These are simply traveling waves in ±x directions so we can simplify
The free particle wavefunction
2 d2ψ The free particle would seem to be the sim- − ~ = Eψ plest case, however, in quantum mechanics 2m dx2 it is actually rather complex and needs to be d2ψ 2m = − Eψ = −k2ψ treated carefully. dx2 ~2 The free particle Hamiltonian has V ≡ 0 everywhere
Carlo Segre (Illinois Tech) PHYS 405 - Fundamentals of Quantum Theory I The free particle √ 2mE ≡ Aei(kx−ωt), k = ± ~
ψ(x) = Aeikx + Be−ikx
Adding the time-dependent portion e−i(E/~)t = e−iωt
Ψ(x, t) = Aei(kx−ωt) + Be−i(kx+ωt)
These are simply traveling waves in ±x directions so we can simplify
The free particle wavefunction
2 d2ψ The free particle would seem to be the sim- − ~ = Eψ plest case, however, in quantum mechanics 2m dx2 it is actually rather complex and needs to be d2ψ 2m = − Eψ = −k2ψ treated carefully. dx2 ~2 The free particle Hamiltonian has V ≡ 0 everywhere In this case, however, our trial solution needs to be of the form
Carlo Segre (Illinois Tech) PHYS 405 - Fundamentals of Quantum Theory I The free particle √ 2mE ≡ Aei(kx−ωt), k = ± ~
Adding the time-dependent portion e−i(E/~)t = e−iωt
Ψ(x, t) = Aei(kx−ωt) + Be−i(kx+ωt)
These are simply traveling waves in ±x directions so we can simplify
The free particle wavefunction
2 d2ψ The free particle would seem to be the sim- − ~ = Eψ plest case, however, in quantum mechanics 2m dx2 it is actually rather complex and needs to be d2ψ 2m = − Eψ = −k2ψ treated carefully. dx2 ~2 The free particle Hamiltonian has V ≡ 0 everywhere In this case, however, our trial solution needs to be of the form ψ(x) = Aeikx + Be−ikx
Carlo Segre (Illinois Tech) PHYS 405 - Fundamentals of Quantum Theory I The free particle √ 2mE ≡ Aei(kx−ωt), k = ± ~
Ψ(x, t) = Aei(kx−ωt) + Be−i(kx+ωt)
These are simply traveling waves in ±x directions so we can simplify
The free particle wavefunction
2 d2ψ The free particle would seem to be the sim- − ~ = Eψ plest case, however, in quantum mechanics 2m dx2 it is actually rather complex and needs to be d2ψ 2m = − Eψ = −k2ψ treated carefully. dx2 ~2 The free particle Hamiltonian has V ≡ 0 everywhere In this case, however, our trial solution needs to be of the form ψ(x) = Aeikx + Be−ikx
Adding the time-dependent portion e−i(E/~)t = e−iωt
Carlo Segre (Illinois Tech) PHYS 405 - Fundamentals of Quantum Theory I The free particle √ 2mE ≡ Aei(kx−ωt), k = ± ~ These are simply traveling waves in ±x directions so we can simplify
The free particle wavefunction
2 d2ψ The free particle would seem to be the sim- − ~ = Eψ plest case, however, in quantum mechanics 2m dx2 it is actually rather complex and needs to be d2ψ 2m = − Eψ = −k2ψ treated carefully. dx2 ~2 The free particle Hamiltonian has V ≡ 0 everywhere In this case, however, our trial solution needs to be of the form ψ(x) = Aeikx + Be−ikx
Adding the time-dependent portion e−i(E/~)t = e−iωt
Ψ(x, t) = Aei(kx−ωt) + Be−i(kx+ωt)
Carlo Segre (Illinois Tech) PHYS 405 - Fundamentals of Quantum Theory I The free particle √ 2mE ≡ Aei(kx−ωt), k = ± ~ so we can simplify
The free particle wavefunction
2 d2ψ The free particle would seem to be the sim- − ~ = Eψ plest case, however, in quantum mechanics 2m dx2 it is actually rather complex and needs to be d2ψ 2m = − Eψ = −k2ψ treated carefully. dx2 ~2 The free particle Hamiltonian has V ≡ 0 everywhere In this case, however, our trial solution needs to be of the form ψ(x) = Aeikx + Be−ikx
Adding the time-dependent portion e−i(E/~)t = e−iωt
Ψ(x, t) = Aei(kx−ωt) + Be−i(kx+ωt)
These are simply traveling waves in ±x directions Carlo Segre (Illinois Tech) PHYS 405 - Fundamentals of Quantum Theory I The free particle The free particle wavefunction
2 d2ψ The free particle would seem to be the sim- − ~ = Eψ plest case, however, in quantum mechanics 2m dx2 it is actually rather complex and needs to be d2ψ 2m = − Eψ = −k2ψ treated carefully. dx2 ~2 The free particle Hamiltonian has V ≡ 0 everywhere In this case, however, our trial solution needs to be of the form ψ(x) = Aeikx + Be−ikx
Adding the time-dependent portion e−i(E/~)t = e−iωt √ 2mE Ψ(x, t) = Aei(kx−ωt) + Be−i(kx+ωt) ≡ Aei(kx−ωt), k = ± ~ These are simply traveling waves in ±x directions so we can simplify Carlo Segre (Illinois Tech) PHYS 405 - Fundamentals of Quantum Theory I The free particle the wavelength of this particle is λ = 2π/|k|
the momentum, according to the de Broglie relation is p = ~k
the speed of the traveling waves described ω k2 1 v = = ~ by this wave function is qm |k| 2m |k| √ r ~|k| ~ 2mE E this is half the classical speed coming from = = = 1 2 2m 2m ~ 2m E = 2 mvcl
r the wavefunction Ψ(x, t) cannot be normal- 2E v = = 2v ized as it does not trend to zero at ±∞ cl m qm Such a state with definite energy is not a valid state for a free particle!
Properties of the free particle
Ψ(x, t) = Aei(kx−ωt) √ 2mE k = ± ~
Carlo Segre (Illinois Tech) PHYS 405 - Fundamentals of Quantum Theory I The free particle the momentum, according to the de Broglie relation is p = ~k
the speed of the traveling waves described ω k2 1 v = = ~ by this wave function is qm |k| 2m |k| √ r ~|k| ~ 2mE E this is half the classical speed coming from = = = 1 2 2m 2m ~ 2m E = 2 mvcl
r the wavefunction Ψ(x, t) cannot be normal- 2E v = = 2v ized as it does not trend to zero at ±∞ cl m qm Such a state with definite energy is not a valid state for a free particle!
Properties of the free particle
Ψ(x, t) = Aei(kx−ωt) the wavelength of this particle is λ = 2π/|k| √ 2mE k = ± ~
Carlo Segre (Illinois Tech) PHYS 405 - Fundamentals of Quantum Theory I The free particle the speed of the traveling waves described ω k2 1 v = = ~ by this wave function is qm |k| 2m |k| √ r ~|k| ~ 2mE E this is half the classical speed coming from = = = 1 2 2m 2m ~ 2m E = 2 mvcl
r the wavefunction Ψ(x, t) cannot be normal- 2E v = = 2v ized as it does not trend to zero at ±∞ cl m qm Such a state with definite energy is not a valid state for a free particle!
Properties of the free particle
Ψ(x, t) = Aei(kx−ωt) the wavelength of this particle is λ = 2π/|k| √ 2mE k = ± the momentum, according to the de Broglie ~ relation is p = ~k
Carlo Segre (Illinois Tech) PHYS 405 - Fundamentals of Quantum Theory I The free particle ω k2 1 v = = ~ qm |k| 2m |k| √ r ~|k| ~ 2mE E this is half the classical speed coming from = = = 1 2 2m 2m ~ 2m E = 2 mvcl
r the wavefunction Ψ(x, t) cannot be normal- 2E v = = 2v ized as it does not trend to zero at ±∞ cl m qm Such a state with definite energy is not a valid state for a free particle!
Properties of the free particle
Ψ(x, t) = Aei(kx−ωt) the wavelength of this particle is λ = 2π/|k| √ 2mE k = ± the momentum, according to the de Broglie ~ relation is p = ~k
the speed of the traveling waves described by this wave function is
Carlo Segre (Illinois Tech) PHYS 405 - Fundamentals of Quantum Theory I The free particle k2 1 = ~ 2m |k| √ r ~|k| ~ 2mE E this is half the classical speed coming from = = = 1 2 2m 2m ~ 2m E = 2 mvcl
r the wavefunction Ψ(x, t) cannot be normal- 2E v = = 2v ized as it does not trend to zero at ±∞ cl m qm Such a state with definite energy is not a valid state for a free particle!
Properties of the free particle
Ψ(x, t) = Aei(kx−ωt) the wavelength of this particle is λ = 2π/|k| √ 2mE k = ± the momentum, according to the de Broglie ~ relation is p = ~k
the speed of the traveling waves described ω v = by this wave function is qm |k|
Carlo Segre (Illinois Tech) PHYS 405 - Fundamentals of Quantum Theory I The free particle √ r ~|k| ~ 2mE E this is half the classical speed coming from = = = 1 2 2m 2m ~ 2m E = 2 mvcl
r the wavefunction Ψ(x, t) cannot be normal- 2E v = = 2v ized as it does not trend to zero at ±∞ cl m qm Such a state with definite energy is not a valid state for a free particle!
Properties of the free particle
Ψ(x, t) = Aei(kx−ωt) the wavelength of this particle is λ = 2π/|k| √ 2mE k = ± the momentum, according to the de Broglie ~ relation is p = ~k
the speed of the traveling waves described ω k2 1 v = = ~ by this wave function is qm |k| 2m |k|
Carlo Segre (Illinois Tech) PHYS 405 - Fundamentals of Quantum Theory I The free particle √ r ~ 2mE E this is half the classical speed coming from = = 1 2 2m ~ 2m E = 2 mvcl
r the wavefunction Ψ(x, t) cannot be normal- 2E v = = 2v ized as it does not trend to zero at ±∞ cl m qm Such a state with definite energy is not a valid state for a free particle!
Properties of the free particle
Ψ(x, t) = Aei(kx−ωt) the wavelength of this particle is λ = 2π/|k| √ 2mE k = ± the momentum, according to the de Broglie ~ relation is p = ~k
the speed of the traveling waves described ω k2 1 v = = ~ by this wave function is qm |k| 2m |k| |k| = ~ 2m
Carlo Segre (Illinois Tech) PHYS 405 - Fundamentals of Quantum Theory I The free particle r E this is half the classical speed coming from = 1 2 2m E = 2 mvcl
r the wavefunction Ψ(x, t) cannot be normal- 2E v = = 2v ized as it does not trend to zero at ±∞ cl m qm Such a state with definite energy is not a valid state for a free particle!
Properties of the free particle
Ψ(x, t) = Aei(kx−ωt) the wavelength of this particle is λ = 2π/|k| √ 2mE k = ± the momentum, according to the de Broglie ~ relation is p = ~k
the speed of the traveling waves described ω k2 1 v = = ~ by this wave function is qm |k| 2m |k| √ |k| 2mE = ~ = ~ 2m 2m ~
Carlo Segre (Illinois Tech) PHYS 405 - Fundamentals of Quantum Theory I The free particle this is half the classical speed coming from 1 2 E = 2 mvcl
r the wavefunction Ψ(x, t) cannot be normal- 2E v = = 2v ized as it does not trend to zero at ±∞ cl m qm Such a state with definite energy is not a valid state for a free particle!
Properties of the free particle
Ψ(x, t) = Aei(kx−ωt) the wavelength of this particle is λ = 2π/|k| √ 2mE k = ± the momentum, according to the de Broglie ~ relation is p = ~k
the speed of the traveling waves described ω k2 1 v = = ~ by this wave function is qm |k| 2m |k| √ r |k| 2mE E = ~ = ~ = 2m 2m ~ 2m
Carlo Segre (Illinois Tech) PHYS 405 - Fundamentals of Quantum Theory I The free particle r the wavefunction Ψ(x, t) cannot be normal- 2E v = = 2v ized as it does not trend to zero at ±∞ cl m qm Such a state with definite energy is not a valid state for a free particle!
Properties of the free particle
Ψ(x, t) = Aei(kx−ωt) the wavelength of this particle is λ = 2π/|k| √ 2mE k = ± the momentum, according to the de Broglie ~ relation is p = ~k
the speed of the traveling waves described ω k2 1 v = = ~ by this wave function is qm |k| 2m |k| √ r ~|k| ~ 2mE E this is half the classical speed coming from = = = 1 2 2m 2m ~ 2m E = 2 mvcl
Carlo Segre (Illinois Tech) PHYS 405 - Fundamentals of Quantum Theory I The free particle the wavefunction Ψ(x, t) cannot be normal- ized as it does not trend to zero at ±∞
Such a state with definite energy is not a valid state for a free particle!
Properties of the free particle
Ψ(x, t) = Aei(kx−ωt) the wavelength of this particle is λ = 2π/|k| √ 2mE k = ± the momentum, according to the de Broglie ~ relation is p = ~k
the speed of the traveling waves described ω k2 1 v = = ~ by this wave function is qm |k| 2m |k| √ r ~|k| ~ 2mE E this is half the classical speed coming from = = = 1 2 2m 2m ~ 2m E = 2 mvcl
r 2E v = = 2v cl m qm
Carlo Segre (Illinois Tech) PHYS 405 - Fundamentals of Quantum Theory I The free particle Such a state with definite energy is not a valid state for a free particle!
Properties of the free particle
Ψ(x, t) = Aei(kx−ωt) the wavelength of this particle is λ = 2π/|k| √ 2mE k = ± the momentum, according to the de Broglie ~ relation is p = ~k
the speed of the traveling waves described ω k2 1 v = = ~ by this wave function is qm |k| 2m |k| √ r ~|k| ~ 2mE E this is half the classical speed coming from = = = 1 2 2m 2m ~ 2m E = 2 mvcl
r the wavefunction Ψ(x, t) cannot be normal- 2E v = = 2v ized as it does not trend to zero at ±∞ cl m qm
Carlo Segre (Illinois Tech) PHYS 405 - Fundamentals of Quantum Theory I The free particle Properties of the free particle
Ψ(x, t) = Aei(kx−ωt) the wavelength of this particle is λ = 2π/|k| √ 2mE k = ± the momentum, according to the de Broglie ~ relation is p = ~k
the speed of the traveling waves described ω k2 1 v = = ~ by this wave function is qm |k| 2m |k| √ r ~|k| ~ 2mE E this is half the classical speed coming from = = = 1 2 2m 2m ~ 2m E = 2 mvcl
r the wavefunction Ψ(x, t) cannot be normal- 2E v = = 2v ized as it does not trend to zero at ±∞ cl m qm Such a state with definite energy is not a valid state for a free particle!
Carlo Segre (Illinois Tech) PHYS 405 - Fundamentals of Quantum Theory I The free particle 1 Z +∞ Ψ(x, t) = √ φ(k)ei(kx−ωt)dk 2π −∞
1 Z +∞ φ(k) = √ Ψ(x, 0)e−ikx dx This momentum distribution is simply 2π −∞ the Fourier transform of the t = 0 Recall the relationship between Fourier transforms wavefunction Ψ(x, 0).
1 Z +∞ 1 Z +∞ f (x) = √ F (k)eikx dk ⇐⇒ F (k) = √ f (x)e−ikx dx 2π −∞ 2π −∞
The real free particle
While these stationary states are not possible wavefunctions themselves, they can be used to construct a valid free particle wavefunction, called a wave packet with a momentum dis- tribution φ(k).
Carlo Segre (Illinois Tech) PHYS 405 - Fundamentals of Quantum Theory I The free particle 1 Z +∞ φ(k) = √ Ψ(x, 0)e−ikx dx This momentum distribution is simply 2π −∞ the Fourier transform of the t = 0 Recall the relationship between Fourier transforms wavefunction Ψ(x, 0).
1 Z +∞ 1 Z +∞ f (x) = √ F (k)eikx dk ⇐⇒ F (k) = √ f (x)e−ikx dx 2π −∞ 2π −∞
The real free particle
While these stationary states are not possible wavefunctions themselves, 1 Z +∞ they can be used to construct a valid Ψ(x, t) = √ φ(k)ei(kx−ωt)dk free particle wavefunction, called a 2π −∞ wave packet with a momentum dis- tribution φ(k).
Carlo Segre (Illinois Tech) PHYS 405 - Fundamentals of Quantum Theory I The free particle 1 Z +∞ φ(k) = √ Ψ(x, 0)e−ikx dx 2π −∞ Recall the relationship between Fourier transforms
1 Z +∞ 1 Z +∞ f (x) = √ F (k)eikx dk ⇐⇒ F (k) = √ f (x)e−ikx dx 2π −∞ 2π −∞
The real free particle
While these stationary states are not possible wavefunctions themselves, 1 Z +∞ they can be used to construct a valid Ψ(x, t) = √ φ(k)ei(kx−ωt)dk free particle wavefunction, called a 2π −∞ wave packet with a momentum dis- tribution φ(k). This momentum distribution is simply the Fourier transform of the t = 0 wavefunction Ψ(x, 0).
Carlo Segre (Illinois Tech) PHYS 405 - Fundamentals of Quantum Theory I The free particle Recall the relationship between Fourier transforms
1 Z +∞ 1 Z +∞ f (x) = √ F (k)eikx dk ⇐⇒ F (k) = √ f (x)e−ikx dx 2π −∞ 2π −∞
The real free particle
While these stationary states are not possible wavefunctions themselves, 1 Z +∞ they can be used to construct a valid Ψ(x, t) = √ φ(k)ei(kx−ωt)dk free particle wavefunction, called a 2π −∞ wave packet with a momentum dis- tribution φ(k). 1 Z +∞ φ(k) = √ Ψ(x, 0)e−ikx dx This momentum distribution is simply 2π −∞ the Fourier transform of the t = 0 wavefunction Ψ(x, 0).
Carlo Segre (Illinois Tech) PHYS 405 - Fundamentals of Quantum Theory I The free particle 1 Z +∞ 1 Z +∞ f (x) = √ F (k)eikx dk ⇐⇒ F (k) = √ f (x)e−ikx dx 2π −∞ 2π −∞
The real free particle
While these stationary states are not possible wavefunctions themselves, 1 Z +∞ they can be used to construct a valid Ψ(x, t) = √ φ(k)ei(kx−ωt)dk free particle wavefunction, called a 2π −∞ wave packet with a momentum dis- tribution φ(k). 1 Z +∞ φ(k) = √ Ψ(x, 0)e−ikx dx This momentum distribution is simply 2π −∞ the Fourier transform of the t = 0 Recall the relationship between Fourier transforms wavefunction Ψ(x, 0).
Carlo Segre (Illinois Tech) PHYS 405 - Fundamentals of Quantum Theory I The free particle 1 Z +∞ ⇐⇒ F (k) = √ f (x)e−ikx dx 2π −∞
The real free particle
While these stationary states are not possible wavefunctions themselves, 1 Z +∞ they can be used to construct a valid Ψ(x, t) = √ φ(k)ei(kx−ωt)dk free particle wavefunction, called a 2π −∞ wave packet with a momentum dis- tribution φ(k). 1 Z +∞ φ(k) = √ Ψ(x, 0)e−ikx dx This momentum distribution is simply 2π −∞ the Fourier transform of the t = 0 Recall the relationship between Fourier transforms wavefunction Ψ(x, 0).
1 Z +∞ f (x) = √ F (k)eikx dk 2π −∞
Carlo Segre (Illinois Tech) PHYS 405 - Fundamentals of Quantum Theory I The free particle The real free particle
While these stationary states are not possible wavefunctions themselves, 1 Z +∞ they can be used to construct a valid Ψ(x, t) = √ φ(k)ei(kx−ωt)dk free particle wavefunction, called a 2π −∞ wave packet with a momentum dis- tribution φ(k). 1 Z +∞ φ(k) = √ Ψ(x, 0)e−ikx dx This momentum distribution is simply 2π −∞ the Fourier transform of the t = 0 Recall the relationship between Fourier transforms wavefunction Ψ(x, 0).
1 Z +∞ 1 Z +∞ f (x) = √ F (k)eikx dk ⇐⇒ F (k) = √ f (x)e−ikx dx 2π −∞ 2π −∞
Carlo Segre (Illinois Tech) PHYS 405 - Fundamentals of Quantum Theory I The free particle r E v = qm 2m r 2E vcl = = 2vqm The “quantum” velocity and the “classical” velocity m
The wave packet is a superposition of sinusoidal func- vg tions with amplitude modulated by φ(k). This wave packet looks like a high frequency oscillation modu- vp lated by an envelope
The “quantum” velocity is called the phase veloc- ity, and the “classical” velocity is called the group velocity.
Phase and group velocities
Remember that we computed two different velocities.
Carlo Segre (Illinois Tech) PHYS 405 - Fundamentals of Quantum Theory I The free particle r 2E vcl = = 2vqm and the “classical” velocity m
The wave packet is a superposition of sinusoidal func- vg tions with amplitude modulated by φ(k). This wave packet looks like a high frequency oscillation modu- vp lated by an envelope
The “quantum” velocity is called the phase veloc- ity, and the “classical” velocity is called the group velocity.
Phase and group velocities
r E vqm = Remember that we computed two different velocities. 2m
The “quantum” velocity
Carlo Segre (Illinois Tech) PHYS 405 - Fundamentals of Quantum Theory I The free particle = 2vqm
The wave packet is a superposition of sinusoidal func- vg tions with amplitude modulated by φ(k). This wave packet looks like a high frequency oscillation modu- vp lated by an envelope
The “quantum” velocity is called the phase veloc- ity, and the “classical” velocity is called the group velocity.
Phase and group velocities
r E v = qm 2m Remember that we computed two different velocities. r 2E vcl = The “quantum” velocity and the “classical” velocity m
Carlo Segre (Illinois Tech) PHYS 405 - Fundamentals of Quantum Theory I The free particle The wave packet is a superposition of sinusoidal func- vg tions with amplitude modulated by φ(k). This wave packet looks like a high frequency oscillation modu- vp lated by an envelope
The “quantum” velocity is called the phase veloc- ity, and the “classical” velocity is called the group velocity.
Phase and group velocities
r E v = qm 2m Remember that we computed two different velocities. r 2E vcl = = 2vqm The “quantum” velocity and the “classical” velocity m
Carlo Segre (Illinois Tech) PHYS 405 - Fundamentals of Quantum Theory I The free particle vg This wave packet looks like a high frequency oscillation modu- vp lated by an envelope
The “quantum” velocity is called the phase veloc- ity, and the “classical” velocity is called the group velocity.
Phase and group velocities
r E v = qm 2m Remember that we computed two different velocities. r 2E vcl = = 2vqm The “quantum” velocity and the “classical” velocity m
The wave packet is a superposition of sinusoidal func- tions with amplitude modulated by φ(k).
Carlo Segre (Illinois Tech) PHYS 405 - Fundamentals of Quantum Theory I The free particle The “quantum” velocity is called the phase veloc- ity, and the “classical” velocity is called the group velocity.
Phase and group velocities
r E v = qm 2m Remember that we computed two different velocities. r 2E vcl = = 2vqm The “quantum” velocity and the “classical” velocity m
The wave packet is a superposition of sinusoidal func- vg tions with amplitude modulated by φ(k). This wave packet looks like a high frequency oscillation modu- vp lated by an envelope
Carlo Segre (Illinois Tech) PHYS 405 - Fundamentals of Quantum Theory I The free particle Phase and group velocities
r E v = qm 2m Remember that we computed two different velocities. r 2E vcl = = 2vqm The “quantum” velocity and the “classical” velocity m
The wave packet is a superposition of sinusoidal func- vg tions with amplitude modulated by φ(k). This wave packet looks like a high frequency oscillation modu- vp lated by an envelope
The “quantum” velocity is called the phase veloc- ity, and the “classical” velocity is called the group velocity.
Carlo Segre (Illinois Tech) PHYS 405 - Fundamentals of Quantum Theory I The free particle Let’s determine the velocity of the free particle wave packet, starting with the definition: 1 Z +∞ Ψ(x, t) = √ φ(k)ei(kx −ωt)dk 2π −∞
assuming that φ(k) is a narrow distribution about k0, only values of k ≈ k0 contribute to the integral ∼ 0 approximate ω(k) with the first two terms ω(k) = ω0 + ω0(k − k0) ∼ 0 of its Taylor expansion. = ω0 + ω0s
change variables to s ≡ k − k0 k = k0 + s, dk = ds +∞ 1 Z 0 ∼ i[(k0+s)x−(ω0+ω s)t] Ψ(x, t) = √ φ(k0 + s)e 0 ds 2π −∞
Wave packet velocity
The relationship between the group and phase velocities depends on the physical system and is called the “dispersion relation” and is just the expression ω(k), the angular frequency as a function of the wave number.
Carlo Segre (Illinois Tech) PHYS 405 - Fundamentals of Quantum Theory I The free particle 1 Z +∞ Ψ(x, t) = √ φ(k)ei(kx −ωt)dk 2π −∞
assuming that φ(k) is a narrow distribution about k0, only values of k ≈ k0 contribute to the integral ∼ 0 approximate ω(k) with the first two terms ω(k) = ω0 + ω0(k − k0) ∼ 0 of its Taylor expansion. = ω0 + ω0s
change variables to s ≡ k − k0 k = k0 + s, dk = ds +∞ 1 Z 0 ∼ i[(k0+s)x−(ω0+ω s)t] Ψ(x, t) = √ φ(k0 + s)e 0 ds 2π −∞
Wave packet velocity
The relationship between the group and phase velocities depends on the physical system and is called the “dispersion relation” and is just the expression ω(k), the angular frequency as a function of the wave number. Let’s determine the velocity of the free particle wave packet, starting with the definition:
Carlo Segre (Illinois Tech) PHYS 405 - Fundamentals of Quantum Theory I The free particle assuming that φ(k) is a narrow distribution about k0, only values of k ≈ k0 contribute to the integral ∼ 0 approximate ω(k) with the first two terms ω(k) = ω0 + ω0(k − k0) ∼ 0 of its Taylor expansion. = ω0 + ω0s
change variables to s ≡ k − k0 k = k0 + s, dk = ds +∞ 1 Z 0 ∼ i[(k0+s)x−(ω0+ω s)t] Ψ(x, t) = √ φ(k0 + s)e 0 ds 2π −∞
Wave packet velocity
The relationship between the group and phase velocities depends on the physical system and is called the “dispersion relation” and is just the expression ω(k), the angular frequency as a function of the wave number. Let’s determine the velocity of the free particle wave packet, starting with the definition: 1 Z +∞ Ψ(x, t) = √ φ(k)ei(kx−ωt)dk 2π −∞
Carlo Segre (Illinois Tech) PHYS 405 - Fundamentals of Quantum Theory I The free particle ∼ 0 approximate ω(k) with the first two terms ω(k) = ω0 + ω0(k − k0) ∼ 0 of its Taylor expansion. = ω0 + ω0s
change variables to s ≡ k − k0 k = k0 + s, dk = ds +∞ 1 Z 0 ∼ i[(k0+s)x−(ω0+ω s)t] Ψ(x, t) = √ φ(k0 + s)e 0 ds 2π −∞
Wave packet velocity
The relationship between the group and phase velocities depends on the physical system and is called the “dispersion relation” and is just the expression ω(k), the angular frequency as a function of the wave number. Let’s determine the velocity of the free particle wave packet, starting with the definition: 1 Z +∞ Ψ(x, t) = √ φ(k)ei(kx−ωt)dk 2π −∞
assuming that φ(k) is a narrow distribution about k0, only values of k ≈ k0 contribute to the integral
Carlo Segre (Illinois Tech) PHYS 405 - Fundamentals of Quantum Theory I The free particle ∼ 0 ω(k) = ω0 + ω0(k − k0) ∼ 0 = ω0 + ω0s
change variables to s ≡ k − k0 k = k0 + s, dk = ds +∞ 1 Z 0 ∼ i[(k0+s)x−(ω0+ω s)t] Ψ(x, t) = √ φ(k0 + s)e 0 ds 2π −∞
Wave packet velocity
The relationship between the group and phase velocities depends on the physical system and is called the “dispersion relation” and is just the expression ω(k), the angular frequency as a function of the wave number. Let’s determine the velocity of the free particle wave packet, starting with the definition: 1 Z +∞ Ψ(x, t) = √ φ(k)ei(kx−ωt)dk 2π −∞
assuming that φ(k) is a narrow distribution about k0, only values of k ≈ k0 contribute to the integral approximate ω(k) with the first two terms of its Taylor expansion.
Carlo Segre (Illinois Tech) PHYS 405 - Fundamentals of Quantum Theory I The free particle ∼ 0 = ω0 + ω0s
change variables to s ≡ k − k0 k = k0 + s, dk = ds +∞ 1 Z 0 ∼ i[(k0+s)x−(ω0+ω s)t] Ψ(x, t) = √ φ(k0 + s)e 0 ds 2π −∞
Wave packet velocity
The relationship between the group and phase velocities depends on the physical system and is called the “dispersion relation” and is just the expression ω(k), the angular frequency as a function of the wave number. Let’s determine the velocity of the free particle wave packet, starting with the definition: 1 Z +∞ Ψ(x, t) = √ φ(k)ei(kx−ωt)dk 2π −∞
assuming that φ(k) is a narrow distribution about k0, only values of k ≈ k0 contribute to the integral ∼ 0 approximate ω(k) with the first two terms ω(k) = ω0 + ω0(k − k0) of its Taylor expansion.
Carlo Segre (Illinois Tech) PHYS 405 - Fundamentals of Quantum Theory I The free particle ∼ 0 = ω0 + ω0s k = k0 + s, dk = ds +∞ 1 Z 0 ∼ i[(k0+s)x−(ω0+ω s)t] Ψ(x, t) = √ φ(k0 + s)e 0 ds 2π −∞
Wave packet velocity
The relationship between the group and phase velocities depends on the physical system and is called the “dispersion relation” and is just the expression ω(k), the angular frequency as a function of the wave number. Let’s determine the velocity of the free particle wave packet, starting with the definition: 1 Z +∞ Ψ(x, t) = √ φ(k)ei(kx−ωt)dk 2π −∞
assuming that φ(k) is a narrow distribution about k0, only values of k ≈ k0 contribute to the integral ∼ 0 approximate ω(k) with the first two terms ω(k) = ω0 + ω0(k − k0) of its Taylor expansion.
change variables to s ≡ k − k0
Carlo Segre (Illinois Tech) PHYS 405 - Fundamentals of Quantum Theory I The free particle ∼ 0 = ω0 + ω0s
+∞ 1 Z 0 ∼ i[(k0+s)x−(ω0+ω s)t] Ψ(x, t) = √ φ(k0 + s)e 0 ds 2π −∞
Wave packet velocity
The relationship between the group and phase velocities depends on the physical system and is called the “dispersion relation” and is just the expression ω(k), the angular frequency as a function of the wave number. Let’s determine the velocity of the free particle wave packet, starting with the definition: 1 Z +∞ Ψ(x, t) = √ φ(k)ei(kx−ωt)dk 2π −∞
assuming that φ(k) is a narrow distribution about k0, only values of k ≈ k0 contribute to the integral ∼ 0 approximate ω(k) with the first two terms ω(k) = ω0 + ω0(k − k0) of its Taylor expansion.
change variables to s ≡ k − k0 k = k0 + s, dk = ds
Carlo Segre (Illinois Tech) PHYS 405 - Fundamentals of Quantum Theory I The free particle +∞ 1 Z 0 ∼ i[(k0+s)x−(ω0+ω s)t] Ψ(x, t) = √ φ(k0 + s)e 0 ds 2π −∞
Wave packet velocity
The relationship between the group and phase velocities depends on the physical system and is called the “dispersion relation” and is just the expression ω(k), the angular frequency as a function of the wave number. Let’s determine the velocity of the free particle wave packet, starting with the definition: 1 Z +∞ Ψ(x, t) = √ φ(k)ei(kx−ωt)dk 2π −∞
assuming that φ(k) is a narrow distribution about k0, only values of k ≈ k0 contribute to the integral ∼ 0 approximate ω(k) with the first two terms ω(k) = ω0 + ω0(k − k0) ∼ 0 of its Taylor expansion. = ω0 + ω0s
change variables to s ≡ k − k0 k = k0 + s, dk = ds
Carlo Segre (Illinois Tech) PHYS 405 - Fundamentals of Quantum Theory I The free particle +∞ 1 Z 0 ∼ i[(k0+s)x−(ω0+ω s)t] Ψ(x, t) = √ φ(k0 + s)e 0 ds 2π −∞
Wave packet velocity
The relationship between the group and phase velocities depends on the physical system and is called the “dispersion relation” and is just the expression ω(k), the angular frequency as a function of the wave number. Let’s determine the velocity of the free particle wave packet, starting with the definition: 1 Z +∞ Ψ(x, t) = √ φ(k)ei(kx −ωt)dk 2π −∞
assuming that φ(k) is a narrow distribution about k0, only values of k ≈ k0 contribute to the integral ∼ 0 approximate ω(k) with the first two terms ω(k) = ω0 + ω0(k − k0) ∼ 0 of its Taylor expansion. = ω0 + ω0s
change variables to s ≡ k − k0 k = k0 + s, dk = ds
Carlo Segre (Illinois Tech) PHYS 405 - Fundamentals of Quantum Theory I The free particle +∞ 1 Z 0 ∼ i[(k0+s)x−(ω0+ω s)t] Ψ(x, t) = √ φ(k0 + s)e 0 ds 2π −∞
Wave packet velocity
The relationship between the group and phase velocities depends on the physical system and is called the “dispersion relation” and is just the expression ω(k), the angular frequency as a function of the wave number. Let’s determine the velocity of the free particle wave packet, starting with the definition: 1 Z +∞ Ψ(x, t) = √ φ(k)ei(kx −ωt)dk 2π −∞
assuming that φ(k) is a narrow distribution about k0, only values of k ≈ k0 contribute to the integral ∼ 0 approximate ω(k) with the first two terms ω(k) = ω0 + ω0(k − k0) ∼ 0 of its Taylor expansion. = ω0 + ω0s
change variables to s ≡ k − k0 k = k0 + s, dk = ds
Carlo Segre (Illinois Tech) PHYS 405 - Fundamentals of Quantum Theory I The free particle +∞ 1 Z 0 ∼ i[(k0+s)x−(ω0+ω s)t] Ψ(x, t) = √ φ(k0 + s)e 0 ds 2π −∞
Wave packet velocity
The relationship between the group and phase velocities depends on the physical system and is called the “dispersion relation” and is just the expression ω(k), the angular frequency as a function of the wave number. Let’s determine the velocity of the free particle wave packet, starting with the definition: 1 Z +∞ Ψ(x, t) = √ φ(k)ei(kx −ωt)dk 2π −∞
assuming that φ(k) is a narrow distribution about k0, only values of k ≈ k0 contribute to the integral ∼ 0 approximate ω(k) with the first two terms ω(k) = ω0 + ω0(k − k0) ∼ 0 of its Taylor expansion. = ω0 + ω0s
change variables to s ≡ k − k0 k = k0 + s, dk = ds
Carlo Segre (Illinois Tech) PHYS 405 - Fundamentals of Quantum Theory I The free particle Wave packet velocity
The relationship between the group and phase velocities depends on the physical system and is called the “dispersion relation” and is just the expression ω(k), the angular frequency as a function of the wave number. Let’s determine the velocity of the free particle wave packet, starting with the definition: 1 Z +∞ Ψ(x, t) = √ φ(k)ei(kx −ωt)dk 2π −∞
assuming that φ(k) is a narrow distribution about k0, only values of k ≈ k0 contribute to the integral ∼ 0 approximate ω(k) with the first two terms ω(k) = ω0 + ω0(k − k0) ∼ 0 of its Taylor expansion. = ω0 + ω0s
change variables to s ≡ k − k0 k = k0 + s, dk = ds +∞ 1 Z 0 ∼ i[(k0+s)x−(ω0+ω s)t] Ψ(x, t) = √ φ(k0 + s)e 0 ds 2π −∞
Carlo Segre (Illinois Tech) PHYS 405 - Fundamentals of Quantum Theory I The free particle 0 and with x → x − ω0t
at t = 0 this is exactly and at time t it can be rewritten 1 Z +∞ i(k0+s)x Ψ(x, 0) = √ φ(k0 + s)e ds 2π −∞ +∞ 1 Z 0 0 0 ∼ i[(k0+s)x−(ω0+ω s)t−k0ω t] ik0ω t Ψ(x, t) = √ φ(k0 + s)e 0 0 e 0 ds 2π −∞ +∞ 1 0 Z 0 ∼ i(−ω0t+k0ω t) i(k0+s)(x−ω t) = √ e 0 φ(k0 + s)e 0 ds 2π −∞ the equation at t is identical to that at t = 0 except with a phase shift
0 ∼ −i(ω0−k0ω0)t 0 Ψ(x, t) = e Ψ(x − ω0t, 0)
in a time t the wave packet has moved a distance ω0 t 0 dω 0 vg = ω = 0 dk implying that k0
Wave packet velocity +∞ 1 Z 0 ∼ i[(k0+s)x−(ω0+ω s)t] Ψ(x, t) = √ φ(k0 + s)e 0 ds 2π −∞
Carlo Segre (Illinois Tech) PHYS 405 - Fundamentals of Quantum Theory I The free particle 0 and with x → x − ω0t
and at time t it can be rewritten 1 Z +∞ i(k0+s)x Ψ(x, 0) = √ φ(k0 + s)e ds 2π −∞ +∞ 1 Z 0 0 0 ∼ i[(k0+s)x−(ω0+ω s)t−k0ω t] ik0ω t Ψ(x, t) = √ φ(k0 + s)e 0 0 e 0 ds 2π −∞ +∞ 1 0 Z 0 ∼ i(−ω0t+k0ω t) i(k0+s)(x−ω t) = √ e 0 φ(k0 + s)e 0 ds 2π −∞ the equation at t is identical to that at t = 0 except with a phase shift
0 ∼ −i(ω0−k0ω0)t 0 Ψ(x, t) = e Ψ(x − ω0t, 0)
in a time t the wave packet has moved a distance ω0 t 0 dω 0 vg = ω = 0 dk implying that k0
Wave packet velocity +∞ 1 Z 0 ∼ i[(k0+s)x−(ω0+ω s)t] Ψ(x, t) = √ φ(k0 + s)e 0 ds 2π −∞ at t = 0 this is exactly
Carlo Segre (Illinois Tech) PHYS 405 - Fundamentals of Quantum Theory I The free particle 0 and with x → x − ω0t
and at time t it can be rewritten
+∞ 1 Z 0 0 0 ∼ i[(k0+s)x−(ω0+ω s)t−k0ω t] ik0ω t Ψ(x, t) = √ φ(k0 + s)e 0 0 e 0 ds 2π −∞ +∞ 1 0 Z 0 ∼ i(−ω0t+k0ω t) i(k0+s)(x−ω t) = √ e 0 φ(k0 + s)e 0 ds 2π −∞ the equation at t is identical to that at t = 0 except with a phase shift
0 ∼ −i(ω0−k0ω0)t 0 Ψ(x, t) = e Ψ(x − ω0t, 0)
in a time t the wave packet has moved a distance ω0 t 0 dω 0 vg = ω = 0 dk implying that k0
Wave packet velocity +∞ 1 Z 0 ∼ i[(k0+s)x−(ω0+ω s)t] Ψ(x, t) = √ φ(k0 + s)e 0 ds 2π −∞ at t = 0 this is exactly 1 Z +∞ i(k0+s)x Ψ(x, 0) = √ φ(k0 + s)e ds 2π −∞
Carlo Segre (Illinois Tech) PHYS 405 - Fundamentals of Quantum Theory I The free particle 0 and with x → x − ω0t
+∞ 1 Z 0 0 0 ∼ i[(k0+s)x−(ω0+ω s)t−k0ω t] ik0ω t Ψ(x, t) = √ φ(k0 + s)e 0 0 e 0 ds 2π −∞ +∞ 1 0 Z 0 ∼ i(−ω0t+k0ω t) i(k0+s)(x−ω t) = √ e 0 φ(k0 + s)e 0 ds 2π −∞ the equation at t is identical to that at t = 0 except with a phase shift
0 ∼ −i(ω0−k0ω0)t 0 Ψ(x, t) = e Ψ(x − ω0t, 0)
in a time t the wave packet has moved a distance ω0 t 0 dω 0 vg = ω = 0 dk implying that k0
Wave packet velocity +∞ 1 Z 0 ∼ i[(k0+s)x−(ω0+ω s)t] Ψ(x, t) = √ φ(k0 + s)e 0 ds 2π −∞ at t = 0 this is exactly and at time t it can be rewritten 1 Z +∞ i(k0+s)x Ψ(x, 0) = √ φ(k0 + s)e ds 2π −∞
Carlo Segre (Illinois Tech) PHYS 405 - Fundamentals of Quantum Theory I The free particle 0 and with x → x − ω0t
+∞ 1 0 Z 0 ∼ i(−ω0t+k0ω t) i(k0+s)(x−ω t) = √ e 0 φ(k0 + s)e 0 ds 2π −∞ the equation at t is identical to that at t = 0 except with a phase shift
0 ∼ −i(ω0−k0ω0)t 0 Ψ(x, t) = e Ψ(x − ω0t, 0)
in a time t the wave packet has moved a distance ω0 t 0 dω 0 vg = ω = 0 dk implying that k0
Wave packet velocity +∞ 1 Z 0 ∼ i[(k0+s)x−(ω0+ω s)t] Ψ(x, t) = √ φ(k0 + s)e 0 ds 2π −∞ at t = 0 this is exactly and at time t it can be rewritten 1 Z +∞ i(k0+s)x Ψ(x, 0) = √ φ(k0 + s)e ds 2π −∞ +∞ 1 Z 0 0 0 ∼ i[(k0+s)x−(ω0+ω s)t−k0ω t] ik0ω t Ψ(x, t) = √ φ(k0 + s)e 0 0 e 0 ds 2π −∞
Carlo Segre (Illinois Tech) PHYS 405 - Fundamentals of Quantum Theory I The free particle 0 and with x → x − ω0t
the equation at t is identical to that at t = 0 except with a phase shift
0 ∼ −i(ω0−k0ω0)t 0 Ψ(x, t) = e Ψ(x − ω0t, 0)
in a time t the wave packet has moved a distance ω0 t 0 dω 0 vg = ω = 0 dk implying that k0
Wave packet velocity +∞ 1 Z 0 ∼ i[(k0+s)x−(ω0+ω s)t] Ψ(x, t) = √ φ(k0 + s)e 0 ds 2π −∞ at t = 0 this is exactly and at time t it can be rewritten 1 Z +∞ i(k0+s)x Ψ(x, 0) = √ φ(k0 + s)e ds 2π −∞ +∞ 1 Z 0 0 0 ∼ i[(k0+s)x−(ω0+ω s)t−k0ω t] ik0ω t Ψ(x, t) = √ φ(k0 + s)e 0 0 e 0 ds 2π −∞ +∞ 1 0 Z 0 ∼ i(−ω0t+k0ω t) i(k0+s)(x−ω t) = √ e 0 φ(k0 + s)e 0 ds 2π −∞
Carlo Segre (Illinois Tech) PHYS 405 - Fundamentals of Quantum Theory I The free particle 0 and with x → x − ω0t 0 ∼ −i(ω0−k0ω0)t 0 Ψ(x, t) = e Ψ(x − ω0t, 0)
in a time t the wave packet has moved a distance ω0 t 0 dω 0 vg = ω = 0 dk implying that k0
Wave packet velocity +∞ 1 Z 0 ∼ i[(k0+s)x−(ω0+ω s)t] Ψ(x, t) = √ φ(k0 + s)e 0 ds 2π −∞ at t = 0 this is exactly and at time t it can be rewritten 1 Z +∞ i(k0+s)x Ψ(x, 0) = √ φ(k0 + s)e ds 2π −∞ +∞ 1 Z 0 0 0 ∼ i[(k0+s)x−(ω0+ω s)t−k0ω t] ik0ω t Ψ(x, t) = √ φ(k0 + s)e 0 0 e 0 ds 2π −∞ +∞ 1 0 Z 0 ∼ i(−ω0t+k0ω t) i(k0+s)(x−ω t) = √ e 0 φ(k0 + s)e 0 ds 2π −∞ the equation at t is identical to that at t = 0 except with a phase shift
Carlo Segre (Illinois Tech) PHYS 405 - Fundamentals of Quantum Theory I The free particle 0 ∼ −i(ω0−k0ω0)t 0 Ψ(x, t) = e Ψ(x − ω0t, 0)
in a time t the wave packet has moved a distance ω0 t 0 dω 0 vg = ω = 0 dk implying that k0
Wave packet velocity +∞ 1 Z 0 ∼ i[(k0+s)x−(ω0+ω s)t] Ψ(x, t) = √ φ(k0 + s)e 0 ds 2π −∞ at t = 0 this is exactly and at time t it can be rewritten 1 Z +∞ i(k0+s)x Ψ(x, 0) = √ φ(k0 + s)e ds 2π −∞ +∞ 1 Z 0 0 0 ∼ i[(k0+s)x−(ω0+ω s)t−k0ω t] ik0ω t Ψ(x, t) = √ φ(k0 + s)e 0 0 e 0 ds 2π −∞ +∞ 1 0 Z 0 ∼ i(−ω0t+k0ω t) i(k0+s)(x−ω t) = √ e 0 φ(k0 + s)e 0 ds 2π −∞ 0 the equation at t is identical to that at t = 0 except with a phase shift and with x → x − ω0t
Carlo Segre (Illinois Tech) PHYS 405 - Fundamentals of Quantum Theory I The free particle
in a time t the wave packet has moved a distance ω0 t 0 dω 0 vg = ω = 0 dk implying that k0
Wave packet velocity +∞ 1 Z 0 ∼ i[(k0+s)x−(ω0+ω s)t] Ψ(x, t) = √ φ(k0 + s)e 0 ds 2π −∞ at t = 0 this is exactly and at time t it can be rewritten 1 Z +∞ i(k0+s)x Ψ(x, 0) = √ φ(k0 + s)e ds 2π −∞ +∞ 1 Z 0 0 0 ∼ i[(k0+s)x−(ω0+ω s)t−k0ω t] ik0ω t Ψ(x, t) = √ φ(k0 + s)e 0 0 e 0 ds 2π −∞ +∞ 1 0 Z 0 ∼ i(−ω0t+k0ω t) i(k0+s)(x−ω t) = √ e 0 φ(k0 + s)e 0 ds 2π −∞ 0 the equation at t is identical to that at t = 0 except with a phase shift and with x → x − ω0t 0 ∼ −i(ω0−k0ω0)t 0 Ψ(x, t) = e Ψ(x − ω0t, 0)
Carlo Segre (Illinois Tech) PHYS 405 - Fundamentals of Quantum Theory I The free particle
0 dω vg = ω = 0 dk k0
Wave packet velocity +∞ 1 Z 0 ∼ i[(k0+s)x−(ω0+ω s)t] Ψ(x, t) = √ φ(k0 + s)e 0 ds 2π −∞ at t = 0 this is exactly and at time t it can be rewritten 1 Z +∞ i(k0+s)x Ψ(x, 0) = √ φ(k0 + s)e ds 2π −∞ +∞ 1 Z 0 0 0 ∼ i[(k0+s)x−(ω0+ω s)t−k0ω t] ik0ω t Ψ(x, t) = √ φ(k0 + s)e 0 0 e 0 ds 2π −∞ +∞ 1 0 Z 0 ∼ i(−ω0t+k0ω t) i(k0+s)(x−ω t) = √ e 0 φ(k0 + s)e 0 ds 2π −∞ 0 the equation at t is identical to that at t = 0 except with a phase shift and with x → x − ω0t 0 ∼ −i(ω0−k0ω0)t 0 Ψ(x, t) = e Ψ(x − ω0t, 0) 0 in a time t the wave packet has moved a distance ω0t implying that Carlo Segre (Illinois Tech) PHYS 405 - Fundamentals of Quantum Theory I The free particle
dω = dk k0
Wave packet velocity +∞ 1 Z 0 ∼ i[(k0+s)x−(ω0+ω s)t] Ψ(x, t) = √ φ(k0 + s)e 0 ds 2π −∞ at t = 0 this is exactly and at time t it can be rewritten 1 Z +∞ i(k0+s)x Ψ(x, 0) = √ φ(k0 + s)e ds 2π −∞ +∞ 1 Z 0 0 0 ∼ i[(k0+s)x−(ω0+ω s)t−k0ω t] ik0ω t Ψ(x, t) = √ φ(k0 + s)e 0 0 e 0 ds 2π −∞ +∞ 1 0 Z 0 ∼ i(−ω0t+k0ω t) i(k0+s)(x−ω t) = √ e 0 φ(k0 + s)e 0 ds 2π −∞ 0 the equation at t is identical to that at t = 0 except with a phase shift and with x → x − ω0t 0 ∼ −i(ω0−k0ω0)t 0 Ψ(x, t) = e Ψ(x − ω0t, 0) 0 in a time t the wave packet has moved a distance ω0t 0 vg = ω0 implying that Carlo Segre (Illinois Tech) PHYS 405 - Fundamentals of Quantum Theory I The free particle Wave packet velocity +∞ 1 Z 0 ∼ i[(k0+s)x−(ω0+ω s)t] Ψ(x, t) = √ φ(k0 + s)e 0 ds 2π −∞ at t = 0 this is exactly and at time t it can be rewritten 1 Z +∞ i(k0+s)x Ψ(x, 0) = √ φ(k0 + s)e ds 2π −∞ +∞ 1 Z 0 0 0 ∼ i[(k0+s)x−(ω0+ω s)t−k0ω t] ik0ω t Ψ(x, t) = √ φ(k0 + s)e 0 0 e 0 ds 2π −∞ +∞ 1 0 Z 0 ∼ i(−ω0t+k0ω t) i(k0+s)(x−ω t) = √ e 0 φ(k0 + s)e 0 ds 2π −∞ 0 the equation at t is identical to that at t = 0 except with a phase shift and with x → x − ω0t 0 ∼ −i(ω0−k0ω0)t 0 Ψ(x, t) = e Ψ(x − ω0t, 0)
in a time t the wave packet has moved a distance ω0 t 0 dω 0 vg = ω = 0 dk implying that k0 Carlo Segre (Illinois Tech) PHYS 405 - Fundamentals of Quantum Theory I The free particle
dω vg = dk and the k0 ω velocity of individual wave function compo- vp = nents k
k2 since the relation between the frequency and ω = ~ wave number (the dispersion relation) is 2m
k v = ~ 0 = v the two velocities become g m cl the group velocity is twice the velocity of the ~k ~k0 vp = → = vqm phase velocity of the central component of 2m 2m the momentum distribution this is evident in the simulation
Free particle velocities
For a free particle, the two velocities are thus, the velocity of the wave packet
Carlo Segre (Illinois Tech) PHYS 405 - Fundamentals of Quantum Theory I The free particle and the ω velocity of individual wave function compo- vp = nents k
k2 since the relation between the frequency and ω = ~ wave number (the dispersion relation) is 2m
k v = ~ 0 = v the two velocities become g m cl the group velocity is twice the velocity of the ~k ~k0 vp = → = vqm phase velocity of the central component of 2m 2m the momentum distribution this is evident in the simulation
Free particle velocities
dω For a free particle, the two velocities are vg = dk thus, the velocity of the wave packet k0
Carlo Segre (Illinois Tech) PHYS 405 - Fundamentals of Quantum Theory I The free particle ω v = p k
k2 since the relation between the frequency and ω = ~ wave number (the dispersion relation) is 2m
k v = ~ 0 = v the two velocities become g m cl the group velocity is twice the velocity of the ~k ~k0 vp = → = vqm phase velocity of the central component of 2m 2m the momentum distribution this is evident in the simulation
Free particle velocities
dω For a free particle, the two velocities are vg = dk thus, the velocity of the wave packet and the k0 velocity of individual wave function compo- nents
Carlo Segre (Illinois Tech) PHYS 405 - Fundamentals of Quantum Theory I The free particle k2 since the relation between the frequency and ω = ~ wave number (the dispersion relation) is 2m
k v = ~ 0 = v the two velocities become g m cl the group velocity is twice the velocity of the ~k ~k0 vp = → = vqm phase velocity of the central component of 2m 2m the momentum distribution this is evident in the simulation
Free particle velocities
dω For a free particle, the two velocities are vg = dk thus, the velocity of the wave packet and the k0 ω velocity of individual wave function compo- vp = nents k
Carlo Segre (Illinois Tech) PHYS 405 - Fundamentals of Quantum Theory I The free particle k2 ω = ~ 2m
k v = ~ 0 = v the two velocities become g m cl the group velocity is twice the velocity of the ~k ~k0 vp = → = vqm phase velocity of the central component of 2m 2m the momentum distribution this is evident in the simulation
Free particle velocities
dω For a free particle, the two velocities are vg = dk thus, the velocity of the wave packet and the k0 ω velocity of individual wave function compo- vp = nents k
since the relation between the frequency and wave number (the dispersion relation) is
Carlo Segre (Illinois Tech) PHYS 405 - Fundamentals of Quantum Theory I The free particle k v = ~ 0 = v the two velocities become g m cl the group velocity is twice the velocity of the ~k ~k0 vp = → = vqm phase velocity of the central component of 2m 2m the momentum distribution this is evident in the simulation
Free particle velocities
dω For a free particle, the two velocities are vg = dk thus, the velocity of the wave packet and the k0 ω velocity of individual wave function compo- vp = nents k
k2 since the relation between the frequency and ω = ~ wave number (the dispersion relation) is 2m
Carlo Segre (Illinois Tech) PHYS 405 - Fundamentals of Quantum Theory I The free particle k v = ~ 0 = v g m cl the group velocity is twice the velocity of the ~k ~k0 vp = → = vqm phase velocity of the central component of 2m 2m the momentum distribution this is evident in the simulation
Free particle velocities
dω For a free particle, the two velocities are vg = dk thus, the velocity of the wave packet and the k0 ω velocity of individual wave function compo- vp = nents k
k2 since the relation between the frequency and ω = ~ wave number (the dispersion relation) is 2m
the two velocities become
Carlo Segre (Illinois Tech) PHYS 405 - Fundamentals of Quantum Theory I The free particle = vcl
the group velocity is twice the velocity of the ~k ~k0 vp = → = vqm phase velocity of the central component of 2m 2m the momentum distribution this is evident in the simulation
Free particle velocities
dω For a free particle, the two velocities are vg = dk thus, the velocity of the wave packet and the k0 ω velocity of individual wave function compo- vp = nents k
k2 since the relation between the frequency and ω = ~ wave number (the dispersion relation) is 2m
k v = ~ 0 the two velocities become g m
Carlo Segre (Illinois Tech) PHYS 405 - Fundamentals of Quantum Theory I The free particle = vcl
the group velocity is twice the velocity of the ~k0 → = vqm phase velocity of the central component of 2m the momentum distribution this is evident in the simulation
Free particle velocities
dω For a free particle, the two velocities are vg = dk thus, the velocity of the wave packet and the k0 ω velocity of individual wave function compo- vp = nents k
k2 since the relation between the frequency and ω = ~ wave number (the dispersion relation) is 2m
k v = ~ 0 the two velocities become g m k v = ~ p 2m
Carlo Segre (Illinois Tech) PHYS 405 - Fundamentals of Quantum Theory I The free particle = vcl k → ~ 0 = v 2m qm
this is evident in the simulation
Free particle velocities
dω For a free particle, the two velocities are vg = dk thus, the velocity of the wave packet and the k0 ω velocity of individual wave function compo- vp = nents k
k2 since the relation between the frequency and ω = ~ wave number (the dispersion relation) is 2m
k v = ~ 0 the two velocities become g m the group velocity is twice the velocity of the ~k vp = phase velocity of the central component of 2m the momentum distribution
Carlo Segre (Illinois Tech) PHYS 405 - Fundamentals of Quantum Theory I The free particle = vcl
= vqm
this is evident in the simulation
Free particle velocities
dω For a free particle, the two velocities are vg = dk thus, the velocity of the wave packet and the k0 ω velocity of individual wave function compo- vp = nents k
k2 since the relation between the frequency and ω = ~ wave number (the dispersion relation) is 2m
k v = ~ 0 the two velocities become g m the group velocity is twice the velocity of the ~k ~k0 vp = → phase velocity of the central component of 2m 2m the momentum distribution
Carlo Segre (Illinois Tech) PHYS 405 - Fundamentals of Quantum Theory I The free particle Free particle velocities
dω For a free particle, the two velocities are vg = dk thus, the velocity of the wave packet and the k0 ω velocity of individual wave function compo- vp = nents k
k2 since the relation between the frequency and ω = ~ wave number (the dispersion relation) is 2m
k v = ~ 0 = v the two velocities become g m cl the group velocity is twice the velocity of the ~k ~k0 vp = → = vqm phase velocity of the central component of 2m 2m the momentum distribution this is evident in the simulation
Carlo Segre (Illinois Tech) PHYS 405 - Fundamentals of Quantum Theory I The free particle https://scipy-cookbook.readthedocs.io/items/SchrodingerFDTD.html
Free particle wavepacket demo Wavepackets are easier to understand when you visualize them
Carlo Segre (Illinois Tech) PHYS 405 - Fundamentals of Quantum Theory I The free particle https://scipy-cookbook.readthedocs.io/items/SchrodingerFDTD.html
Free particle wavepacket demo Wavepackets are easier to understand when you visualize them
Carlo Segre (Illinois Tech) PHYS 405 - Fundamentals of Quantum Theory I The free particle https://scipy-cookbook.readthedocs.io/items/SchrodingerFDTD.html
Free particle wavepacket demo Wavepackets are easier to understand when you visualize them
Carlo Segre (Illinois Tech) PHYS 405 - Fundamentals of Quantum Theory I The free particle Free particle wavepacket demo Wavepackets are easier to understand when you visualize them
https://scipy-cookbook.readthedocs.io/items/SchrodingerFDTD.html
Carlo Segre (Illinois Tech) PHYS 405 - Fundamentals of Quantum Theory I The free particle Carlo Segre (Illinois Tech) PHYS 405 - Fundamentals of Quantum Theory I The free particle