### Question:The energy eigenvalues of the 1D quantum harmonic oscillator are ________ .1. nondegenerate2. positive3. integral multiples of \(\hbar \omega\)### Options:- ☐ I. and II.- ☐ I.- ☐ II. and III.- ☐ I. and III.

Solution: The Energy eigenvalues of 1D (one -dimensional) quantum harmonic oscillator are discussed below: Explanation: The Energy eigenvalues of 1D (one -dimensional) quantum harmonic oscillator are given as: En=n+12hωWhere,For n=0,1,2....There are countably infinite number of solutions with equal energy spacing The lowest possible energy is: E0=12hω Here, All the eigenvalues are positive. Also if we consider proving the integral multiple of hω are true or not is shown below: Suppose, we have En=n+12hω When n=0,1,2,3...E0=0+12hω=12hωE1=1+12hω=32hωThus, it shows this is not a integral multiple of hω If we take the Value of n=3; then E3=3+12hω=72hωAs per the above eigen value, we only have one eigen function for each value of n.It means that suppose if we take:n1=0 n1=1n2=1 or n2=2n3=2 n3=0Then,En1n2n3=E012=0+1+2+12hω =72hωEn1n2n3=E120=1+2+0+12hω =72hωIn both cases of different wave function values are same with eigen value.Hence, the eigen value has degeneracy but not for all values of nThus,All the energy eigen values are nondegenerate. Conclusion: Thus, we conclude that from the above explanation: The Energy eigenvalues are Nondegenerate and positive. Hence, Option 1 is correct.

Answered: The energy eigenvalues of the 1D…

The energy eigenvalues of the 1D quantum harmonic oscillator are I. nondegenerate II. positive III. integral multiples of ħw O I. and II. OI. O II. and III. O I. and III.