1.3 Diracformulation Vp(x+L.y.z,t)=Vp(x.y+Ly.)=Vp(x.y.+Le.t)Vp(x.y.z.t).resulting in a set of quantized momentum states p=2an(222) 1.11) where n,ny,and n:are integers Control question.Can you derive(1.11)from the periodic boundary conditions and (1.10)yourself? As we see from (1.11),there is a single allowed value of p per volume (h)in momentum space,and therefore the density of momentum states increases with increasing size of the system,D(p)=V/(2h)3.Going toward the limit yoo makes the spac- ing between the discrete momentum values smaller and smaller,and finally results in a continuous spectrum of p. The abov wave e function is one of the simplest solutions of the Schrodinger ation and describesa free particle spread over a volume.nrealhowever functions are usually more complex,and they also can have many components.For exam- ple,if the plane wave in(1.10)describes a single electron,we have to take the spin degree of freedom of the electron into account (see Section 1.7).Since an electron can be in a “spin up”or“spin down”state(or in any superposition of the two).we generally have to use a two-component wave functior (1.12 where the moduli squared of the two components give the relative probabilities to find the electron in the spin up or spin down state as a function of position and time. 1.3 Dirac formulation In the early days of quantum mechanics,the wave function had been thought of as an actual function of space and time coordinates.In this form,it looks very similar to a classical field,i.e.a(multi-component)quantity which is present in every point of coordinate space, such as an electric field E(r,)or a pressure field p(r.).However,it appeared to be very to treat wave functions rathe as elements of a multi-dimensional linear vector space.a Hilbert space.Dirac's formula tion of quantum mechanics enabled a reconciliation of competing approaches to quantum problems and revolutionized the field. In Dirac's approach,every wave function is represented by a vector,which can be put as a"ket"1worba”(1.Ope ators acting on the wave fun tions such as the mentum 7 1.3 Dirac formulation ψp(x + Lx, y,z, t) = ψp(x, y + Ly,z, t) = ψp(x, y,z + Lz, t) = ψp(x, y,z, t), resulting in a set of quantized momentum states p = 2πh¯ nx Lx , ny Ly , nz Lz  , (1.11) where nx, ny, and nz are integers. Control question. Can you derive (1.11) from the periodic boundary conditions and (1.10) yourself? As we see from (1.11), there is a single allowed value of p per volume (2πh¯) 3/V in momentum space, and therefore the density of momentum states increases with increasing size of the system, D(p) = V/(2πh¯) 3. Going toward the limit V → ∞ makes the spac￾ing between the discrete momentum values smaller and smaller, and finally results in a continuous spectrum of p. The above plane wave function is one of the simplest solutions of the Schrödinger equation and describes a free particle spread over a volume V. In reality, however, wave functions are usually more complex, and they also can have many components. For exam￾ple, if the plane wave in (1.10) describes a single electron, we have to take the spin degree of freedom of the electron into account (see Section 1.7). Since an electron can be in a “spin up” or “spin down” state (or in any superposition of the two), we generally have to use a two-component wave function ψp(r, t) =  ψp,↑(r, t) ψp,↓(r, t)  , (1.12) where the moduli squared of the two components give the relative probabilities to find the electron in the spin up or spin down state as a function of position and time. 1.3 Dirac formulation In the early days of quantum mechanics, the wave function had been thought of as an actual function of space and time coordinates. In this form, it looks very similar to a classical field, i.e. a (multi-component) quantity which is present in every point of coordinate space, such as an electric field E(r, t) or a pressure field p(r, t). However, it appeared to be very restrictive to regard a wave function merely as a function of coordinates, as the Schrödinger formalism implies. In his Ph.D. thesis, Paul Dirac proposed to treat wave functions rather as elements of a multi-dimensional linear vector space, a Hilbert space. Dirac’s formula￾tion of quantum mechanics enabled a reconciliation of competing approaches to quantum problems and revolutionized the field. In Dirac’s approach, every wave function is represented by a vector, which can be put as a “ket” |ψ or “bra” ψ|. Operators acting on the wave functions, such as the momentum