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I. Lesson Overview
Syllabus
Defining operators \(a_k, a^\dagger_k,N_k\); Commutation relations and properties of \(a_k, a^\dagger_k,N_k\);
Hilbert space of quantized Schrodinger field; Physical interpretation of states and field operators.
Objectives
- To define operators \(a, a^\dagger, N\) in terms of fields.
- To derive commutation properties of \(N, a ,a^\dagger\)
- To define a complete orthonormal set.
- To construct Hilbert space as linear span of the complete orthonormal set.
- To give interpretation of field operator.
Prerequisites
Quantum Mechanics
Complete orthonormal set
Complete commuting set of hermitian operators
Harmonic oscillator
Vector Spaces
Postulates of quantum mechanics
Simultaneous measurement
Specification of state
II Recall and Discuss
Recall and Discuss --- Vector Spaces
Complete orthonormal set
A set of vectors \(\Osc= \big\{ \ket{\nu}\big| \nu=0,1,2,\ldots\}\) is called
a complete orthonormal set if \\
\begin{equation}
\innerproduct {\nu}{\mu}= \delta_{\mu\nu},
\end{equation}
and if there does not exist another orthonormal set \(\Osc{'}\)
such that \(\Osc \subsetneqq\Osc{'}\).
The importance of complete orthonormal set is that it can be taken as a
basis and the Hilbert space is just the linear span of a complete
othonormal set set.
Complete commuting Set
A set of operators \(\Sca=\big\{A_k\big\}\) is a commuting set if every pair of operators in \(\Sca\) commute.
\[\big[ A_j, A_k\big]=0, \quad \forall j,k\]
The set \(\Sca\) is a complete commuting set if an operator \(Y\) that commutes with all \(X_k\), then \(Y\) is a function of \(X_k\).
The simultaneous eigenvectors of a complete set of commuting hermitian operators forms an orthonormal basis. Every vector in Hilbert space can be written as a unique linear superposition of the simultaneous eigenvectors of complete commuting set of hermitian operators.
Recall and Discuss --- Quantum Mechanics
Postulates of Quantum Mechanics
- States of a Physical system are represented by vectors in a complex vector space with inner product (Hilbert Space).
- Dynamical variables are represented by hermitian operators in the vector space.
- Let \(A\) be dynamical variable and \(\widehat{A}\) be corresponding operator having eigenvalues and eigenvectors \(\alpha_n,\ket{\alpha_n}\). If state is represented by \(\ket{\alpha_k}\) a measurement of \(A\) will give value \(\alpha_k\) with probability 1. Conversely, if measurement of \(A\) gives \(\alpha_k\) with probability 1, then the state is represented by the corresponding eigenvector \(\ket{\alpha_k}\). If the state vector is an arbitrary vector \(\ket{\psi}\), a measurement of \(A\) will give value \(\alpha_k\) with probabilities \(|\innerproduct{\psi}{\alpha_k}|^2\),
Simultaneous measurement
A set of dynamical variables \(A_1, A_2, \ldots\) can be measured
simultaneously if they commute pairwise, {\it i.e.} \([A_j,A_k]=0\) for all pairs \(A_j, A_k\).
State specification
Physically, a complete commuting set of observables can be measured
simultaneously. The probability of all possible outcomes
of such measurement is the maximum information that can be
obtained about the state of the system from experiments.
The state vector can be identified with the set of all such
probability amplitudes.
Harmonic Oscillator Operators $a, a^\dagger,N$.
- The harmonic oscillator creation and annihilation operators obey the commutation rules \begin{equation} \big[a, a^\dagger\big] =1. \end{equation}
- The number operator defined as \(N=a^\dagger a\) obeys the commutation relations \begin{equation} \big[a, N\big]= -a \; \qquad \big[a^\dagger, N\big] =a^\dagger. \end{equation}
- The operator \(a\) and \(a^\dagger\), decrease and increase eigenvalue of \(N\) by one.
- The operator \(N\) is a positive definite operator and has eigenvalues \(\nu=0,1,2,..\).
- The eigenvectors of \(N\) form a complete orthonormal set. The linear span of this set coincides with the Hilbert space.
III. Main Topics Here
Some mathematics of Quantized Schrodinger field
\(\S 1\) Define Operators $a, a^\dagger, N_k$
\(\S 2\) Commutation properties of $a, a^\dagger, N_k$
\(\S 3\) Summary of Properties of $a, a^\dagger, N_k$
\(\S 4\) Constructing the Hilbert Space
\(\S 5\) Physical Interpretation
IV EndNotes
Points of Discussion
More about Harmonic Oscillator Connection
We will now recall and rewrite equations from harmonic oscillator which
show that each \(N_k\) is harmonic oscillator Hamiltonian. For this purpose
we introduce operators \(q_k,p_k\) by
\begin{equation} q_k=\frac{1}{\sqrt{2}}(a_k^\dagger + a_k), \quad p_k=\frac{i}{\sqrt{2}}(a_k^\dagger - a_k) \end{equation}
These operators obey commutation relations
\begin{equation} \big[q_k, q_\ell\big]=\big[p_k, p_\ell\big]=0,\qquad \big[q_k, p_\ell\big]=i\delta_{k\ell}. \end{equation}
The number operators \(N_k\) written in terms of \(q_k,p_k\)take the form
\begin{equation} N_k= \frac{1}{2}\big(q_k^2+ p_k^2\big)-\frac{1}{2} \end{equation}
Thus each \(N_k\), apart from a constant, is like harmonic oscillator Hamiltonian.
Points to Remember
- The simultaneous eigenvectors of \(N_k\) are chosen as an orthonormal basis to get the Hilbert space .
- The simultaneous eigenvectors correspond \(\ket{\nu_1, \nu_2, ..}\) to the state in which there are \(\nu_1, \nu_2 ...\) particles in 'levels' \(u_1,u_2,...\), respectively .
- The physical interpretation of the vectors in the Hilbert space in the number representation depends on the choice of basis functions \(\{u_n(x)\}\) for expansion of field operator.
- Recall the equation of continuity in quantum mechanics \begin{equation} \dd[\rho]{t} + \nabla\cdot {\mathbf J} =0 \end{equation}
- The 'probability density' and the 'probability current' from quantum mechanics now have the following particle interpretation: \(\langle\alpha\rangle\)\(\rho=\psi^\dagger(\mathbf x)\psi(\mathbf x)\) as number of particles per unit volume (\langle\beta\rangle\)\(\mathbf x,t)=\frac{i\hbar}{2m}(\psi^\dagger (\nabla\psi) - (\nabla\psi^\dagger) \psi\) as flux.
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