Chapter 8 Vector Spaces in Quantum Mechanics

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Chapter 8 Vector Spaces in Quantum Mechanics 84Classical vectorQuantum state vector for spinhalf systemBasis Vectors û 1 , û 2 |+〉, |−〉Inner Product (v 1 , v 2 ) = v ∗ 1 · v 2 (|S ′ 〉, |S 〉) = 〈S ′ |S 〉Orthonormality û ∗ 1 · û 1 = û ∗ 2 · û 2 = 1û ∗ 1 · û 2 = û ∗ 2 · û 1 = 0〈+|+〉 = 〈−|−〉 = 1〈+|−〉 = 〈−|+〉 = 0Linearcombinationˆv = aû 1 + bû 2a and b complex numbers|S 〉 = a|+〉 + b|−〉a and b complex numbersNormalisation — 〈S |S 〉 = 1A More Detailed Analysis One of the important properties of vectors is that two ormore of them can be combined as a ‘linear combination’ to produce a third. If we are toconsider quantum states as vectors, then this very basic property must also be possessedby quantum states. In the above, we have not really shown that any linear combination ofthe basis states |±〉, does indeed represent a possible state of the system, even if we restrictourselves to the case of the measuring spin components in the XZ plane. But a moredetailed argument than that just presented can be used to strengthen this conclusion. Tosee how this comes about, we return to the above expression Eq. (7.18) for the probabilityamplitude 〈S f = 1|S 2 i = 1〉:2〈S f = 1 2 |S i = 1 2 〉 =〈S f = 1 2 |S I = 1 2 〉〈S I = 1 2 |S i = 1 2 〉+ 〈S f = 1 2 |S I = − 1 2 〉〈S I = − 1 2 |S i = 1 2 〉. (8.35)We can ‘cancel’ 〈S f = 1 | from this expression and write2|S i = 1〉 = |S 2 I = 1〉〈S 2 I = 1|S 2 i = 1〉 + |S 2 I = − 1〉〈S 2 I = − 1|S 2 i = 1 〉 (8.36)2In Section 7.3, Eq. (7.33), the quantity 〈S f = 1|S 2 i = 1 〉 was shown to be given by2〈S f = 1 2 |S i = 1 2 〉 = cos[(θ f − θ i )/2]with the phase factor exp(iΦ) put equal to unity, and with ˆn and ˆm, the directions of themagnetic fields, confined to the XZ plane. The probability amplitudes for the system topass through the intermediate states |S I = ± 1〉, that is, 〈S 2 I = ± 1|S 2 i = 1 〉 are likewise2given by〈S I = 1 2 |S i = 1 2 〉 = cos[ 1 2 (θ I − θ i )]〈S I = − 1 2 |S i = 1 2 〉 = cos[ 1 2 (θ I + π − θ i )] = − sin[ 1 2 (θ I − θ i )].(8.37)so that Eq. (8.36) becomes|S i = 1〉 = cos[ 1(θ 2 2 I − θ i )]|S I = + 1〉 − sin[ 1(θ 2 2 I − θ i )]|S I = − 1 〉. (8.38)2
Chapter 8 Vector Spaces in Quantum Mechanics 85To further simplify things, we will assume that θ I = 0, i.e. the ˆl vector is in the z direction.In addition, we will make the notational simplification already used above defined by|S i = 1 〉 →|S 〉2|S I = ± 1 〉 →|±〉2〈S I = ± 1|S 2 i = 1 〉 →〈±|S 〉2so that |±〉 = |S z = ± 1 〉, and Eq. (8.36) becomes2or, using Eq. (8.37)|S 〉 = |+〉〈+|S 〉 + |−〉〈−|S 〉. (8.39)|S 〉 = cos( 1 2 θ i)|+〉 + sin( 1 2 θ i)|−〉. (8.40)We are now at the point at which we can begin to supply an interpretation to this equation.What this equation is saying is that the combination cos( 1θ 2 i)|+〉+sin( 1θ 2 i)|−〉, and |S 〉, bothrepresent the same thing – the atomic spin is in a state for which S i = 1 . In other words,2if we were presented with the combination:1√2|+〉 + 1 √2|−〉 (8.41)we immediately see that cos( 1 2 θ i) = 1/ √ 2 and sin( 1 2 θ i) = 1/ √ 2, and hence θ i = 90 ◦ . Thusthe magnetic field is pointing in the direction 90 ◦ to the z direction, i.e. in the x direction,and hence the spin state of the atom is the state |S 〉 = |S x = 1 2 〉.But what if we were presented with the combination 2|+〉 + 2|−〉? Here, we cannot findany angle θ i , so it appears that this combination is not a possible state of the atomic spin.But we can write this as2 √ 1√22[|+〉 + √ 1 ]|−〉(8.42)2which we can now understand as representing 2 √ 2|S x = 1〉. Is 2 √ 2|S2 x = 1 〉 a different2physical state of the system to |S x = 1 〉? Well, it is our notation, so we can say what we2like, and what turns out to be preferable is to say that α|S 〉 describes the same physicalstate as |S 〉, for any value of the constant α. Thus, we can say that 2|+〉+2|−〉 is also a stateof the system, namely 2 √ 2|S x = 1 〉, which represents the same physical information2about the state of the system as |S x = 1〉.2Thus any combination C + |+〉 + C − |−〉 where C ± are real numbers will always representsome state of the system, in general given by√C+ 2 + C− 2 |S i = 1 〉 (8.43)2whereS i = S · ˆn (8.44)and where ˆn is a unit vector in the direction defined by the angleθ i = 2 tan −1 (C−C +). (8.45)
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