CH202 Final: The energy representation
The energy representation
Suppose our system is a particle that is trapped in some potential
well. Then the spectrum of allowed energies will be a set of discrete
numbers E0 , E1 , . . . and a complete set of amplitudes are the
amplitudes ai whose mod squares give the probabilities pi of
measuring the energy to be Ei. Let {|iۧ} be a set of basis kets for the
space V of the system’s quantum states. Then we use the set of
amplitudes ai to associate them with a ket |ψۧ through
|ψۧ = ai|iۧ. (1.28) i
This equation relates a complete set of amplitudes {ai} to a certain
ket |ψۧ. We discover the physical meaning of a particular basis ket,
say |kۧ, by examining the values that the expansion coefficients ai
take when we apply equation (1.28) in the case |kۧ = |ψۧ. We clearly
then have that ai = 0 for i ̸= k and ak = 1. Consequently, the quantum
state |kۧ is that in which we are certain to measure the value Ek for
the energy. We say that |kۧ is a state of well defined energy. It will
help us remember this important identification if we relabel the
basis
kets, writing |Eiۧ instead of just |iۧ, so that (1.28) becomes
|ψۧ = ai|Eiۧ. (1.29) i
Suppose we multiply this equation through by ۦEk|. Then by the lin-
earity of this operation and the orthogonality relation (1.22) (which
in our new notation reads ۦEk|Eiۧ = δik) we find
ak = ۦEk|ψۧ. (1.30)
This is an enormously important result because it tells us how to
extract from an arbitrary quantum state |ψۧ the amplitude for
finding that the energy is Ek.
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Document Summary
This equation relates a complete set of amplitudes {ai} to a certain help us remember this important identification if we relabel the ai|i(cid:1767). (1. 28) i. Suppose our system is a particle that is trapped in some potential well. Let {|i(cid:1767)} be a set of basis kets for the space v of the system"s quantum states. Then we use the set of amplitudes ai to associate them with a ket | (cid:1767) through. We discover the physical meaning of a particular basis ket, say |k(cid:1767), by examining the values that the expansion coefficients ai take when we apply equation (1. 28) in the case |k(cid:1767) = | (cid:1767). We clearly then have that ai = 0 for i = k and ak = 1. Consequently, the quantum state |k(cid:1767) is that in which we are certain to measure the value ek for the energy. We say that |k(cid:1767) is a state of well defined energy.
