5. Suppose a particle of mass m, in a tube of length L, is...

5. Suppose a particle of mass m, in a tube of length L, is in the ground state. The tube is suddenly stretched to twice its length, by moving one end of the tube to a new position at x = 2L. Immediately after the tube is stretched, the wave function of the particle is the same as before (it takes time for a wave function to change). If we now measure the energy of the particle, what energy is most likely to be found?

Transcript

00:01 Hi, in this question, we have been given psi of x is equal to a times x e raised to the power minus x squared divided by a squared.

00:14 Now if we differentiate partially with respect to x, then we will get this.

00:21 So here, first of all, we will keep a times e raised to the power minus x squared divided by a squared constant and we will differentiate this x.

00:29 So this will be 1, then plus.

00:31 Now we will keep a times x constant and we will just differentiate this e raised to the power minus x squared divided by a squared.

00:40 So this will be e raised to the power minus x squared divided by a squared multiplied by minus 2x divided by a squared.

00:50 Now we can write this as a e raised to the power minus x squared divided by a squared.

01:01 And now here this will become 1 minus 2x squared divided by a squared.

01:09 Now this del psi of x with respect to x is equal to a e raised to the power minus x squared divided by a squared multiplied by 1 minus 2x squared divided by a squared.

01:25 Now we need to find the double derivative.

01:28 So again, we will differentiate this with respect to x only.

01:31 So this will be now a e raised to the power minus x squared divided by a squared multiplied by 1 minus 2x squared divided by a squared multiplied by minus 2x divided by a squared plus a e raised to the power minus x squared divided by a squared multiplied by 0 minus 4x divided by a squared...

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