Holooly Plus Logo
Holooly Plus Logo

Question 39.57: An electron is trapped in a one-dimensional infinite potenti......

An electron is trapped in a one-dimensional infinite potential well. Show that the energy difference ΔE between its quantum levels n and n + 2 is (h²/2mL²)(n + 1).

Step-by-Step
The 'Blue Check Mark' means that this solution was answered by an expert.
Learn more on how do we answer questions.
Report Answer

From Eq. 39-4,

E_n=\left(\frac{h^2}{8 m L^2}\right) n^2, \quad \text { for } n=1,2,3, \ldots             (39-4)

E_{n+2}-E_n=\left(\frac{h^2}{8 m L^2}\right) ( n+2 )^2-\left(\frac{h^2}{8 m L^2}\right) n^2=\left(\frac{h^2}{2 m L^2}\right) ( n+1 ) .

Related Answered Questions

Question: 39.62

(a) What is the wavelength of light for the least energetic photon emitted in the Balmer series of the hydrogen atom spectrum lines? (b) What is the wavelength of the series limit? ...

Verified Answer:

(a) The “home-base” energy level for the Balmer se...
Question: 39.60

An electron is confined to a narrow evacuated tube of length 3.0 m; the tube functions as a one-dimensional infinite potential well. (a) What is the energy difference between the electron’s ground state and its first excited state? (b) At what quantum number n would the energy difference between ...

Verified Answer:

We can use the mc² value for an electron from Tabl...
Question: 39.56

Let ΔEadj be the energy difference between two adjacent energy levels for an electron trapped in a one-dimensional infinite potential well. Let E be the energy of either of the two levels. (a) Show that the ratio ΔEadj/E approaches the value 2/n at large values of the quantum number n. As n → ∞ , ...

Verified Answer:

(a) The allowed energy values are given by ...
Question: 39.54

The wave function for the hydrogen-atom quantum state represented by the dot plot shown in Fig. 39-21, which has n = 2 and l = ml = 0, is ψ200(r) = 1/4√2π a^-3/2 (2 – r/a)e^-r/2a ,in which a is the Bohr radius and the subscript on Ψ(r) gives the values of the quantum numbers n, l , ml . (a) Plot ...

Verified Answer:

(a) The plot shown below for \left|\psi_{2...
Question: 39.53

Schrödinger’s equation for states of the hydrogen atom for which the orbital quantum number l is zero is 1/r² d/dr (r²dΨ/dr) + 8π²m/h² [E – U(r)]] Ψ =0. Verify that Eq. 39-39, which describes the ground state of the hydrogen atom, is a solution of this equation. ...

Verified Answer:

THINK The ground state of the hydrogen atom corres...
Question: 39.51

What is the probability that in the ground state of the hydrogen atom, the electron will be found at a radius greater than the Bohr radius? ...

Verified Answer:

According to Sample Problem — “ Probability of det...
Question: 39.46

Calculate the probability that the electron in the hydrogen atom, in its ground state, will be found between spherical shells whose radii are a and 2a, where a is the Bohr radius. ...

Verified Answer:

From Sample Problem — “ Probability of detection o...
Question: 39.44

A hydrogen atom in a state having a binding energy (the energy required to remove an electron) of 0.85 eV makes a transition to a state with an excitation energy (the difference between the energy of the state and that of the ground state) of 10.2 eV. (a) What is the energy of the photon emitted ...

Verified Answer:

(a) Since E_2 = – 0.85 eV and E[lat...
Question: 39.43

In the ground state of the hydrogen atom, the electron has a total energy of -13.6 eV. What are (a) its kinetic energy and (b) its potential energy if the electron is one Bohr radius from the central nucleus? ...

Verified Answer:

(a) and (b) Letting a=5.292 \times 10^{-11...
Question: 39.41

What is the probability that an electron in the ground state of the hydrogen atom will be found between two spherical shells whose radii are r and r + Δr, (a) if r = 0.500a and Δr = 0.010a and (b) if r = 1.00a and Δr = 0.01a, where a is the Bohr radius? (Hint:Δr is small enough to permit the radial ...

Verified Answer:

Since Δr is small, we may calculate the probabilit...