Question

For a particle in the first excited state of harmonic oscillator potential,

a) Calculate 〈?〉1, 〈?〉1, 〈? 2〉1, 〈? 2〉1.

b) Calculate (∆?)1 and (∆?)1.

c) Check the uncertainty principle for this state.

d) Estimate the length of the interval about x=0 which corresponds to the classically allowed domain for the first excited state of harmonic oscillator.

e) Using the result of part (d), show that position uncertainty you get in part (b) is comparable to the classical range of the oscillator for the state n=1.

f) What is the probability of finding the particle outside the classically allowed domain for the state n=1.

Answer #1

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Q. 2(a) For the first excited state of harmonic oscillator,
calculate <x>, <p>, <x2>, and <p2>
explicitly by spatial integration. (b) Check that the
uncertainty principle is obeyed.
(c) Compute <T> and <V>. .

For a particle in the harmonic oscillator potential, calculate
the followings in any stationary state n:
a) 〈?̂?̂〉? b) 〈?̂?̂〉? c) 〈[?̂, ?̂]〉n

Calculate the standard deviation of momentum for the second
excited state of a harmonic oscillator.

A harmonic oscillator with mass m and force constant k is in an
excited state that has quantum number n.
1)
Let pmax=mvmaxx, where vmax is the maximum speed calculated in
the Newtonian analysis of the oscillator. Derive an expression for
pmax in terms of n, ℏ, k, and m.
Express your answer in terms of the variables n, k, m, and the
constant ℏ
2)
Derive an expression for the classical amplitude A in terms of
n, ℏ, k,...

For the simple harmonic oscillator at first excited state the
Hermite polynomial H1=x, (a)find the
normalization constant C1 (b) <x> (c)
<x2>, (d) <V(x)> for this
state.

Calculate the expectation value of momentum for the second
excited state of the harmonic oscillator. Show steps for both
integral and bracket notation please.

Quantum mechanics:
Consider a particle initially in the ground state of the
one-dimensional simple harmonic oscillator. A uniform electric
field is abruptly turned on for a time t and then abruptly turned
off again. What is the probability of transition to the first
excited state?

Consider the ground state of the harmonic oscillator.
(a) (7 pts) Calculate the expectation values <x>,
<x2>, <p> and <p2>.
(b) (3 ps) What is the product ΔxΔp, where the two quantities
are standard deviations? (This is easy if you did part (a)). How
does this answer compare with the prediction of the Uncertainty
Principle?

Assume that the hydrogen molecule behaves exactly like a
harmonic oscillator with a force constant of 573 N/m. (a) Calculate
the energıas, in eV's, of the fundamental and first excited state.
(b) Find the vibrational quantum number that roughly corresponds to
your energy

Find the wave function for the ground state and first two
excited states for a particle in an infinitely deep square well of
width a. Show that the uncertainty relation is
satisfied for position and momentum.

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