Q: For a finite potential well, what happens to the wave function in the classically forbidden region ($$E \u003c U_0$)?

It decays exponentially as $$\psi(x) = \psi_{\text{edge}} e$$^{-(x-L)/\eta}$$ for x$$ \geq L$$

Question 1

Which equation represents the time-independent Schrödinger equation for a particle of mass $$m in a $$potential $$U(x)$$?

Accepted Answer

$$\frac{d^2$\psi}{dx^2$} + \frac{2m}{\$$hbar^2$$}(E - U(x))\psi = 0$$

Question 2

For a particle in a rigid one-dimensional box of length $$L$$, what is the expression for the quantized energy levels?

Accepted Answer

$$E_n = \frac{n^2$ h^2$}{8mL^2$}$$

Question 3

Which boundary condition must the wave function $$\psi(x)$$ satisfy for a particle in a rigid box?

Accepted Answer

$$\psi(0) = 0$$ and $$\psi(L) = 0$$

Question 4

What is the normalized wave function for the $$nth $$quantum state of a particle in a rigid box of length $$L$$?

Accepted Answer

$$\psi_n(x) = \sqrt{\frac{2}{L}} \sin\left(\frac{n\pi x}{L}\right)$$ for $$0 \u003c x \u003c L$$

Question 5

A neutron is confined in a one-dimensional box of length $$L = 10 fm. $$What is the energy of the ground state ($$n=1) if$$ $$m_n = 1.67 	imes 10$$^{-27}$$ kg?

Accepted Answer

$$E_1$$ = \frac{$$h^2$$}{8m_$$nL^2$$}$$

Question 6

Which statement best describes quantum-mechanical tunneling?

Accepted Answer

A particle can pass through an energy barrier even if its energy is less than the barrier height, with no energy loss.

Question 7

For a finite potential well, what happens to the wave function in the classically forbidden region ($$E \u003c U_0$)?

Accepted Answer

It decays exponentially as $$\psi(x) = \psi_{	ext{edge}} e$$^{-(x-L)/\eta}$$ for x$$ \geq L$$

Question 8

What is the penetration distance $$\eta$$ for a particle in a finite potential well?

Accepted Answer

$$\eta = \frac{\hbar}{\sqrt{2m(U_0$ - E)}}$$

Question 9

According to the correspondence principle, what happens as the quantum number $$n$$ becomes very large?

Accepted Answer

The quantum probability density approaches the classical probability density.

Question 10

For a quantum harmonic oscillator, what is the expression for the allowed energy levels?

Accepted Answer

$$E_n = \left(n + \frac{1}{2}\right)\hbar\omega$$

Question 11

Which of the following is NOT a condition that the wave function $$\psi(x)$$ must obey?

Accepted Answer

$$\psi(x)$$ must be infinite at the boundaries.

Question 12

A quantum-well laser operates by trapping electrons in a thin semiconductor layer. What quantum mechanical principle enables the emission of laser light?

Accepted Answer

Electrons make quantum jumps between energy levels, emitting photons.

Question 13

For a particle in a rigid box, why is the ground-state energy $$E_1$$ greater than zero?

Accepted Answer

Heisenberg's uncertainty principle prevents the particle from being at rest.

Question 14

In a finite potential well, why are the allowed energies lower than those for a rigid box of the same width?

Accepted Answer

The wave function extends into the classically forbidden region, lowering the energy.

Question 15

What is the probability that a classical particle in a rigid box of length $$L is $$found in an interval of width $$\delta x$$?

Accepted Answer

$$P = \frac{\delta x}{L}$$

Question 16

Which factor determines the node spacing in the wave function of a bound quantum state?

Accepted Answer

The de Broglie wavelength, which is inversely related to the particle's speed.

Question 17

A scanning tunneling microscope (STM) works by exploiting which quantum mechanical phenomenon?

Accepted Answer

Quantum tunneling of electrons across a narrow gap.

Question 18

For a particle approaching an energy barrier of width $$w$$, what is the probability of tunneling through the barrier?

Accepted Answer

$$P_{	ext{tunnel}} = e$$^{-2w/\eta}$$

Question 19

Which statement about the quantum harmonic oscillator is correct?

Accepted Answer

The energy levels are equally spaced.

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