A particle of mass m is in a region with potential energy operator V = ki. If the particle is in its ground state, then its normalized wavefunction is 2mE Vo(1,t) = (1) and its energy is E, = th, where w = Vk/m. This state is called the ground state of the quantum harmonic oscillator. Recall that the definition of the expectation value of a one-dimensional quantum operator A is (4) = [_ våvdz. (2) where Å is sandwiched betreen the wavefunction and its complex conjugate, NOT in front of both of then. For example, (F) = 0 and (P) = 0, because of the symmetry of the wavefunction about the vertical axis. (a) Calculate the expectation values (P) and () of our Wavefunction. You may find the following Gaussian integral identities useful: (3) dr =0 (4) (5) (b) Standard deviations are defined from expectation values as o, = V() - (2)² and o, = V) - (. Heisenberg's uncertainty principle can be formalized with standard deviations representing uncertainty, o, 2 Show that the ground state of the quantum harmonic oscillator saturates Heisenberg's uncertainty principle, that is, the inequality becomes an equality for Ve(r,f). (c) Show that our wavefunction satisfies E, = E) + (v).
A particle of mass m is in a region with potential energy operator V = ki. If the particle is in its ground state, then its normalized wavefunction is 2mE Vo(1,t) = (1) and its energy is E, = th, where w = Vk/m. This state is called the ground state of the quantum harmonic oscillator. Recall that the definition of the expectation value of a one-dimensional quantum operator A is (4) = [_ våvdz. (2) where Å is sandwiched betreen the wavefunction and its complex conjugate, NOT in front of both of then. For example, (F) = 0 and (P) = 0, because of the symmetry of the wavefunction about the vertical axis. (a) Calculate the expectation values (P) and () of our Wavefunction. You may find the following Gaussian integral identities useful: (3) dr =0 (4) (5) (b) Standard deviations are defined from expectation values as o, = V() - (2)² and o, = V) - (. Heisenberg's uncertainty principle can be formalized with standard deviations representing uncertainty, o, 2 Show that the ground state of the quantum harmonic oscillator saturates Heisenberg's uncertainty principle, that is, the inequality becomes an equality for Ve(r,f). (c) Show that our wavefunction satisfies E, = E) + (v).
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Transcribed Image Text:A particle of mass m is in a region with potential energy operator V = ki. If the
particle is in its ground state, then its normalized wavefunction is
1/4
2mE
Vo(1,t) =
(1)
and its energy is E, = th, where w = Vk/m. This state is called the ground state of
the quantum harmonic oscillator.
Recall that the definition of the expectation value of a one-dimensional quantum
operator A is
(4) = [_ våvdz.
(2)
where Å is sandwiched betreen the wavefunction and its complex conjugate, NOT in
front of both of them. For example, (2) = 0 and () = 0, because of the symmetry of
the wavefunction about the vertical axis.
(a) Calculate the expectation values () and () of our wavefunction. You may find
the following Gaussian integral identities useful:
dr=
(3)
dr =0
(4)
dr=
(5)
(b) Standard deviations are defined from expectation values as o, = V() – (2)²
and o, = V) - (. Heisenberg's uncertainty principle can be formalized with
standard deviations representing uncertainty, o, 2 Show that the ground state of
the quantum harmonic oscillator saturates Heisenberg's uncertainty principle, that is,
the inequality becomes an equality for Ve(r,f).
(c) Show that our wavefunction satisfies E, = E) + (v).
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