I have a few questions I'd like to ask about this example. (C1 was already derived before the second part)
1. What does the line "The rest of the coefficients make up the difference" actually mean?
2. What does "As one might expect...because of the admixture of the excited states" mean?
3. Does the <H> in this case represent the expectation value of total energy in the system?
Questions slightly unrelated to the example above:
4. If a Hamiltonian is written as such: Hψn = Enψn, does it now represent the total allowed energy of the wavefunction with quantum number n?
5. If I derived an equation for ψn(x) in the case of a infinite potential well, and derived an expression for its corresponding coefficient, before putting it all together into a linear combination for the general solution for the time-dependent Schrodinger equation, what exactly am I doing?
2. Homework Equations
The Attempt at a Solution
1 and 2) These lines, to me, suggest that there are other Ψn's in the system apart from that of the ground state. If true, how is this possible? Do excited states not have different "shapes" (different n's different number of nodes)?
If not, what should my interpretation of these statements be?
3 and 4) I have a feeling that H can both describe the total energy in a system and of a particle in a specific state n.
5) My interpretation was that the general solution would also tell me the possible measurement outcomes of energy and their corresponding probabilities. If this is true, is it also true that there is a possibility of measuring more than 1 energy level despite there being only a single particle in the box?
Many, many thanks to anyone who takes the time to help clear up my misconceptions.