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Section 5.3 Quantum Measurements

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As discussed earlier, a quantum state |ϕ⟩ can be written as a linear superposition of a set of states called a basis. As shown before, this basis could be a set of states each representing a specific energy E1, E2, E3, … . The same quantum state |ϕ⟩ could also be written in terms of a basis of states having definite values of some other measurable quantity, say momentum (p1, p2, p3, …). Thus the same state could be written as either

(5.9)|ϕ⟩=c1|E1⟩+c2|E2⟩+c3|E3⟩+⋯(Energy)

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or

(5.10)ϕ⟩=d1|p1⟩+d2|p2⟩+d3|p3⟩+⋯.(Momentum)

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Which linear superposition we use depends upon which quantity we want to measure. If we are interested in doing an experiment in which we measure the energy of the particles, we would therefore use equation (5.9) to compute the probabilities for obtaining the allowed energy values E1,E2,E3,… from the probability amplitudes c1, c2, c3, ….

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In quantum mechanics we consider the measurement of a quantity (say energy E) associated with a particle in the state |ϕ⟩ in the following way, as indicated in Figure 5.3. First, a particle is prepared in some quantum state |ϕ⟩ and then is sent into some device (which I have called an Energy Device in the figure) which is capable of measuring the particle's energy. As a result of the measurement, the device displays the value obtained in the measurement. The value displayed by the device can be only one of the discrete values E1, E2, E3, …, and nothing else! Before the measurement is made, we cannot predict what the resulting value of measured energy will be for a particle in a state such as given by (5.9). We can only calculate the probability of getting a particular value. For instance, Figure 5.3 shows the result of a particular measurement yielding the energy value E2. The probability of obtaining this result is P(E2)=|c2|2.

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Figure 5.3. Conceptual device for measuring the energy of a particle in a state |ϕ⟩. In this example, the result of the measurement is the energy value E2 and the state of the particle collapses to |E2⟩.

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Quantum mechanics also has something new to say about the state of the particle after the measurement is made. In terms of classical physics, the measurement leaves the system untouched and the system is left in the same state as it was before the measurement. On the other hand, quantum mechanics says something completely different:

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Collapse of the State: An ideal measurement of the state of a system forces the system, at the instant of measurement, into a particular basis state vector corresponding to the measured value.

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As indicated in Figure 5.3, if a particle is in the state |ϕ⟩ and a measurement of the energy results in obtaining a value of E2, then the state of the system “collapses” into the state |E2⟩. In general, measurements made on quantum systems affect — and often change — the state of the system. This is a profound statement for it says that the particle being measured and the measurement device can not be treated independently. The measurement device must be considered as part of the system being measured!