State Representation for Quantum Systems

Section 5.2 State Representation for Quantum Systems

In order to proceed further with our discussion of quantum mechanics, we will develop a mathematical framework in which to discuss the more general behavior of quantum systems that will allow us to perform calculations and make predictions that can be compared with experiment. As stated above, the wavefunction formalism gives a description of quantum behavior in which measurement of the position of the particle is of prime importance. A more encompassing mathematical structure for describing quantum behavior was devised by the English theoretical physicist P.A.M. Dirac in 1930 that is capable of describing a wide variety of phenomena, including photons and spin.

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Fundamental to this new description is the concept of the quantum state. Mathematically, the state of a particle is represented by the symbol

| ϕ ⟩

called a “state” or “ket vector.” The quantity

ϕ

written inside the symbol

| \ \ ⟩

usually gives some meaningful description which uniquely describes the state in which the particle is at a certain time. For example, if a particle was in a state of constant momentum

p,

moving uniformly through space not subject to any forces, then we could write

| p ⟩

for its state vector. These state vectors satisfy a set of mathematical rules similar to those for ordinary vectors in 3-dimensional space (like

\vec{A}, \vec{B},

etc.), hence the reason for referring to them as state vectors.

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As an example, we have seen that for a one-dimensional infinite potential square well, a particle placed in this well can have a discrete set of possible energy values (

E_{1},

E_{2},

E_{3},

\dots,

E_{n},

\dots

). For each of these values of energy we can define a state vector (

| E_{1} ⟩,

| E_{2} ⟩,

| E_{3} ⟩,

\dots,

| E_{n} ⟩,

\dots

) that represents the system in a state with that definite value of energy. So for a particle that is placed in the infinite well in a state

| E_{3} ⟩,

if we were to measure the particle's energy we would definitely get the value

E_{3}

as a result of that measurement. The collection of state vectors \{

| E_{1} ⟩,

| E_{2} ⟩,

| E_{3} ⟩,

\dots

\} for all possible states is said to be a basis, (in this case an energy basis). This set of state vectors is analogous to the set of unit vectors \{

\hat{ı},

\hat{ȷ},

\hat{k}

\} that we used with ordinary vectors in three-dimensions. Recall that we could write any 3-dimensional vector

\vec{A}

as a linear combination of these unit vectors

\begin{matrix} (5.1) & \vec{A} = A_{x} \hat{ı} + A_{y} \hat{ȷ} + A_{z} \hat{k} \end{matrix}

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where

A_{x},

A_{y},

and

A_{z}

are the

x

y

-, and

z

-components of

\vec{A} .

In an analogous fashion we can write an expression for an arbitrary state of a quantum system as a linear combination of the energy basis states

\begin{matrix} (5.2) & | ϕ ⟩ = c_{1} | E_{1} ⟩ + c_{2} | E_{2} ⟩ + c_{3} | E_{3} ⟩ + \dots, \end{matrix}

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where the coefficients

c_{1},

c_{2},

\dots,

are complex numbers.¹ In the mathematical structure of quantum mechanics, these coefficients have a special significance when making predictions about the results of measurements on a system. If we were to measure the energy of a particle in a state given by (5.2), then the probability that we will measure a value

E_{3},

for instance, is given by

| c_{3} |^{2} .

That is, the coefficients tell us something about the probability of finding a particle in a certain specific state of the measured quantity, such as energy in this case. It is for this reason that the coefficients

c_{j}

are referred to as probability amplitudes.

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Example 5.1. Superposition of states.

Suppose the possible energies for a certain particle are $E_{1},$ $E_{2},$ $E_{3},$ $E_{4},$ …, with the values of energy such that $E_{1} < E_{2} < E_{3} < E_{4} < \dots .$ Consider a particle in a state given by the following linear superposition of energy basis states:

\begin{matrix} (5.3) & | ϕ ⟩ = \frac{1}{\sqrt{6}} | E_{1} ⟩ + \frac{2 i}{\sqrt{6}} | E_{3} ⟩ + \frac{1}{\sqrt{6}} | E_{4} ⟩ . \end{matrix}

(a) Calculate the probability of obtaining a value $E_{3}$ when the energy of the particle is measured. Calculate similar probabilities for obtaining $E_{1}$ and $E_{4} .$ (b) What is the probability of obtaining a value $E_{2}$ in a measurement of the energy of this particle? (c) What is the probability of obtaining a value of either $E_{1},$ $E_{3},$ or $E_{4}$ when measuring the energy of this particle?

Solution.

(a) To calculate the probability, we determine the coefficient of the basis state vector $| E_{3} ⟩,$ which is the probability amplitude $c_{3} = 2 i / \sqrt{6},$ and calculate its magnitude squared,

\begin{matrix} (5.4) & P (E_{3}) = | c_{3} |^{2} = {| \frac{2 i}{\sqrt{6}} |}^{2} = (\frac{2 i}{\sqrt{6}}) (\frac{- 2 i}{\sqrt{6}}) = \frac{2}{3} . \end{matrix}

In a similar manner, $P (E_{1}) = 1 / 6 = P (E_{4}) .$

(b) Since the basis state vector $| E_{2} ⟩$ does not appear in the linear superposition, this means that the probability amplitude $c_{2} = 0$ and therefore $P (E_{2}) = | c_{2} |^{2} = 0 .$ Therefore we would never obtain the value $E_{2}$ when measuring the energy of the particle in this state.

(c) Since we want to know the probability of measuring any of these values, we simply add the probabilities for measuring each of these values:

\begin{matrix} (5.5) & P (E_{1}, E_{3}, E_{4}) = P (E_{1}) + P (E_{3}) + P (E_{4}) = \frac{1}{6} + \frac{2}{3} + \frac{1}{6} = 1 . \end{matrix}

This makes sense because these are the only three values of energy that could be measured so that the probability of measuring any one of them is 100%.

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In part (c) of the previous example, we see that if the squares of the probability amplitudes in a linear superposition state give us information about probabilities, then the probability amplitudes must satisfy the property

\begin{matrix} (5.6) & | c_{1} |^{2} + | c_{2} |^{2} + | c_{3} |^{2} + | c_{4} |^{2} + \dots = 1 . \end{matrix}

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This condition is referred to as normalization and a state vector that satisfies this condition is said to be normalized. Therefore we conclude:

State vectors which properly describe the quantum state of a particle must be normalized.

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Example 5.2. Normalization of states.

Consider a particle in a state given by the following linear superposition of energy basis states

\begin{matrix} (5.7) & | ϕ ⟩ = \frac{1}{\sqrt{6}} | E_{1} ⟩ + \sqrt{\frac{3}{6}} | E_{2} ⟩ + c_{3} | E_{3} ⟩ . \end{matrix}

Determine some possible values for the probability amplitude $c_{3}$ such that this state is properly normalized.

Solution.

We use the condition given in equation (5.6)

\begin{aligned} | c_{1} |^{2} + | c_{2} |^{2} + | c_{3} |^{2} = & 1 \\ {| \frac{1}{\sqrt{6}} |}^{2} + {| \sqrt{\frac{3}{6}} |}^{2} + | c_{3} |^{2} = & 1 \end{aligned}

which is satisfied for the value $| c_{3} |^{2} = 1 / 3 .$ The simplest solution for the value of the probability amplitude would be $c_{3} = \sqrt{1 / 3} .$ However, the solution $c_{3} = i \sqrt{1 / 3}$ also works because

\begin{matrix} (5.8) & | c_{3} |^{2} = {| i \sqrt{\frac{1}{3}} |}^{2} = (i \sqrt{\frac{1}{3}}) (- i \sqrt{\frac{1}{3}}) = - i^{2} (\frac{1}{3}) = + \frac{1}{3} . \end{matrix}

Recall that a complex number

z

is defined as

z = a + i b,

where

a

and

b

are real numbers and

i

is the imaginary number

i = \sqrt{- 1} .

The “real part” of the complex number is

a,

and

b

is the “imaginary part.” The magnitude of a complex number is

| z | = \sqrt{z z^{*}} = \sqrt{a^{2} + b^{2}},

where

z^{*}

is the complex conjugate as defined in the section Subsection 3.2.2.