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Chapter 2: 2.14P (page 51)

A particle is in the ground state of the harmonic oscillator with classical frequency 蝇, when suddenly the spring constant quadruples, so 蝇'=2蝇, without initially changing the wave function (of course, 唯will now evolve differently, because the Hamiltonian has changed). What is the probability that a measurement of the energy would still return the value 鈩�蝇2? What is the probability of getting 鈩�蝇?

Short Answer

Expert verified

The probability of getting 12鈩�蝇is zero, and the probability of 鈩�蝇is 0.943.

Step by step solution

01

The given values of the question

The particle is in ground state and its classical frequency is 蝇. when the spring becomes entangled in a never-ending tangle 蝇'=2蝇.

02

The probabilities of ℏω2 and ℏω

The new allowed energies are En'=(n+12)鈩�蝇'=2(n+12)蝇=鈩�蝇,3鈩�蝇,5鈩�蝇,....

so the probability of getting12鈩�蝇 is zero.

The probability of getting 鈩�蝇is P0=c02, where c0=鈭�唯(x,0)蠄0'dxwith

Substitute the known values inP0=c02

Thus, the probability of 鈩�蝇is 0.943.

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Most popular questions from this chapter

Find the allowed energies of the half harmonic oscillator

V(x)={(1/2)m蝇2x2,x>0,鈭�,x<0.
(This represents, for example, a spring that can be stretched, but not compressed.) Hint: This requires some careful thought, but very little actual calculation.

a) Compute 鉄�x鉄�, 鉄�p鉄�, 鉄�x2鉄�, 鉄�p2鉄�, for the states 蠄0and蠄1 , by explicit integration. Comment; In this and other problems involving the harmonic oscillator it simplifies matters if you introduce the variable 尉鈮�m蝇鈩�xand the constant 伪鈮�(m蝇蟺鈩�)14.

b) Check the uncertainty principle for these states.

c) Compute 鉄�T鉄�(the average kinetic energy) and 鉄�V鉄� (the average potential energy) for these states. (No new integration allowed). Is their sum what you would expect?

In the ground state of the harmonic oscillator, what is the probability (correct to three significant digits) of finding the particle outside the classically allowed region?

Prove the following three theorem;

a) For normalizable solutions the separation constant E must be real as E0+i蟿and show that if equation 1.20 is to hold for all t,蟿 must be zero.

b) The time - independent wave function localid="1658117146660" 蠄(x) can always be taken to be real, This doesn鈥檛 mean that every solution to the time-independent Schrodinger equation is real; what it says is that if you鈥檝e got one that is not, it can always be expressed as a linear combination of solutions that are . So, you might as well stick to蠄 鈥檚 that are real

c) If is an even function then 蠄(x)can always be taken to be either even or odd

a) Show that the wave function of a particle in the infinite square well returns to its original form after a quantum revival time T = 4ma2/蟺~. That is: 唯 (x, T) = 唯 (x, 0) for any state (not just a stationary state).


(b) What is the classical revival time, for a particle of energy E bouncing back and forth between the walls?


(c) For what energy are the two revival times equal?
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