Quantum Mechanics Lecture Notes PDF

Phys 354 Dr. Wajood Diery  Define Dirac notations. State the orthonormal condition. Normalize a wavefunction.  Chapter 2 Mathematical Tools of Quantum Mechanics Dr.Wajood Diery 3  2.3 Dirac Notation To free state vector (wavefunction) coordinate meaning, D...

Phys 354 Dr. Wajood Diery  Define Dirac notations. State the orthonormal condition. Normalize a wavefunction.  Chapter 2 Mathematical Tools of Quantum Mechanics Dr.Wajood Diery 3  2.3 Dirac Notation To free state vector (wavefunction) coordinate meaning, Dirac introduced the following notation: Complex conjugate Properties of kets, bras and bra-kets If a is a number 4  scalar product     =   (r , t )  (r , t ) d 3r * In one dimension   =   * ( x, t )  ( x, t ) dx Properties of the scalar product *  =  5  The norm 𝜓𝜓 𝜓 𝜓⟩ = real and positive 𝜓 𝜓⟩ = 0only if 𝜓 = 0 𝜓 𝜓⟩ = 1 ⇒ 𝜓 is normalized Orthogonal states Two kets  and  are said to be orthogonal if   = 0 Orthonormal set Two kets 𝜓 and 𝜑 are said to be orthonormal if 𝜓 𝜓⟩ = 1 and 𝜑 𝜑⟩ = 1 𝜓 𝜑⟩ = 0 6  𝜙 𝑎𝑛𝑑 𝜙 𝑎𝑟𝑒 𝑜𝑟𝑡ℎ𝑜𝑛𝑜𝑟𝑚𝑎𝑙 → 𝜙 𝜙 = 1 𝜙 𝜙 =1 𝜙 𝜙 = 𝜙 𝜙 =0 (a) How to∗get the bra? = ⟨𝜓| +⟨𝜒| ⟨𝜓| =(|𝜓⟩) ∗ ⟩ | = (3𝑖 𝜙 − 7𝑖 𝜙 ) ⟩ = −3𝑖⟨𝜙 | + 7𝑖 ⟨𝜙 | − 𝜙 | − 2𝑖⟨𝜙 | = −3𝑖⟨𝜙 |+7𝑖⟨𝜙 | ⟨𝜒| = − 𝜙 | − 2𝑖 ⟨𝜙 | 7 ( )* ( )* = −1 − 3𝑖 𝜙 | + 5𝑖 ⟨𝜙 | (b) ⟨𝜓| = −3𝑖 𝜙 | + 7𝑖 ⟨𝜙 | |𝜒⟩ = − 𝜙 ⟩ + 2𝑖 |𝜙 ⟩ +(−3𝑖)(2𝑖) 𝜙 𝜙 +(7𝑖)(−1) 𝜙 𝜙 +(7𝑖)(2𝑖) 𝜙 𝜙 1 0 0 1 = 3𝑖 − 14 ?? ( 𝜓 𝜒 ⟩)*= (3𝑖 − 14 )* 𝜒 𝜓 = −3𝑖 − 14 ≠ External Example Consider the states |𝜓⟩ = 3 𝜙 ⟩ + 𝜙 ⟩, which is given in terms of two orthonormal eigenstates 𝜙 ⟩, |𝜙 ⟩. Is the sate normalized? If not normalized it ? Solution Before doing any calculations you always have to check if the wavefunction is normalized or not. If it is not normalized, then you have to normalize it. ∗ ( |𝜓⟩)⋆= (3 𝜙 ⟩ + |𝜙 ⟩) Normalization → 𝜓 𝜓 = 1 ⟨𝜓 | = 3 𝜙 | + ⟨𝜙 | 𝜓 𝜓 = (3 𝜙 | + ⟨𝜙 |) ( 3 𝜙 ⟩ + |𝜙 ⟩ ) =9 𝜙 𝜙 +3 𝜙 𝜙 +3 𝜙 𝜙 + 𝜙 𝜙 = 9 + 1 = 10 This is the norm. To normalize the state we divide by the square root of the norm 3 1 |𝜓⟩ = |𝜙 ⟩ + |𝜙 ⟩. 10 10  Define Dirac notations. State the orthornormal condition. Normalize a wavefunction.